I 


C-NRLF 


SB    33    3DD 


LIBRARY 

OF    THE 

UNIVERSITY  OF  CALIFORNIA. 
Glass 


iel 


•- :. 


WORKS  OF  PROF.  A.  P.  JAMISON 

PUBLISHED    BY 

JOHN  WILEY  &  SONS. 


Elements  of  Mechanical  Drawing. 

Their  Application  and  A  Course  in  Mechanical 
Drawing  for  Engineering  Students.  8vo,  xii  -f 
226  pages,  including  57  full-page  plates  and  82  fig- 
ures in  the  text.  Cloth,  $2.50. 

Advanced  Mechanical  Drawing. 

A  Text  for  Engineering  Students.  8vo,  ix  +  i77 
pages,  including  27  full-page  plates  and  117  figures 
in  the  text.  Cloth,  $2.00. 


ADVANCED 


MECHANICAL   DRAWING 


A  TEXT  FOR  ENGINEERING  STUDENTS 


BY 


ALPHA    PIERCE   JAMISON,    M.E. 

Assistant  Professor  of  Mechanical  Drawing  in  Purdn«  University 


FIRST    EDITION 
FIRST  THOUSAND 


Of  THE 

UNIVERSITY 

OF 


NEW  YORK 
JOHN  WILEY  &  SONS 


LONDON:    CHAPMAN   &    HALL,    LIMITED 
1905 


Copyright,  1905 

BY 
ALPHA  PIERCE  JAMISON 


ROBERT  DRUMMOND,    PRINTER,    NEW    YORK 


PREFACE. 


Having  in  charge  the  preparation  of  all  of  the  engineering 
students  in  Purdue  University  in  Mechanical  Drawing  for  their 
Course  in  Engineering  Design,  the  writer  has  compiled  a  series 
of  progressive  notes  on  the  subject  calculated  to  impart  a  working 
knowledge  of  the  principles  of  graphic  representation,  and  offer- 
ing such  examples  as  will  acquaint  the  student  with  the  con- 
ventions of  the  art.  The  work  is  divided  into  two  parts,  Part  I 
being  "A  Course  in  Elementary  Mechanical  Drawing,"  ad- 
ministered in  the  Freshman  year,  and  Part  II  a  course  in  "Ad- 
vanced Mechanical  Drawing,"  administered  in  the  Sophomore 
year  as  a  course  in  drawing,  and  in  connection  with  the  class- 
room and  lecture  work  in  Descriptive  Geometry. 

The  work  is  purely  elementary,  dealing  with  methods  of 
representation  alone,  manipulations  of  construction,  and  does 
not  treat  of  Design,  being  preliminary  to  that  subject. 

This  part,  Advanced  Drawing,  is  offered  to  students  and 
draughtsmen  who  have  a  working  knowledge  of  the  principles 
of  the  art,  such  as  is  offered  in  Part  I,  and  who  have,  also,  some 
knowledge  of  the  principles  of  Descriptive  Geometry. 

The  discussions  have  been  made  as  brief  as  was  thought 
consistent  with  clearness,  and  are  intended  simply  to  suggest 
such  lines  of  thought  as  will  render  the  figures,  the  illustrations— 
an  engineer's  "description" — self-explanatory. 

In  selecting  a  "Course  in  Drawing"  from  the  examples 
offered,  it  is  suggested  that,  in  so  far  as  possible,  the  Practical 

iii 


iv  PREFACE. 

Problems  be  made  to  follow  the  Theoretical  Problems  delineat- 
ing the  principle  involved;  such  an  arrangement,  for  example, 
as  is  given  by  Plate  27,  page  177. 

The  writer  has  enjoyed  the  advice  and  co-operation  of  Prof. 
M.  J.  Golden,  Mr.  A.  M.  Wilson,  Mr.  E.  B,  Smith,  and  Mr.  O. 
E.  Williams  in  the  preparation  of  the  manuscript  and  illustra- 
tions, and  wishes  to  thank  them  for  their  many  courtesies  and 
valued  assistance. 

A.  P.  JAMISON. 

LA  FAYETTE,  IND.,  May,  1905. 


CONTENTS. 


PART  I. 
THEORY,  DEFINITIONS,  ETC. 

CHAPTER  I. 

ISOMETRIC  DRAWING  AND  CAVALIER  PROJECTION. 
ISOMETRIC  DRAWING. 

SECTION  PACK 

1 .  Definition i 

2.  Theory 2 

3.  Explanation  of  Terms 3 

Origin 3 

Isometric  Axes 3 

Isometric  Planes 3 

Isometric  Lines 3 

4.  Distinction  between  Isometric  Drawing  and  Isometric  Projection 3 

5.  Isometric  Scales.  ...                                          4 

6.  Practical  Application  of  the  Theory 5 

7.  Method  of  Procedure 6 

8.  Flexibility 8 

9.  Practical  Examples 8 

Plane  Figures 8 

A  Square 8 

A  Circle 10 

A  Triangle n 

A  Hexagon 12 

An  Octagon.  ...                    13 

A  Star 14 

An  Ellipse 14 

A  Parabola 14 

Any  Irregular  Figure 14 

Circular  Arcs 14 

T 


vi  CONTENTS. 

SECTION  PAGE 

Curves 17 

A  Helix 17 

Solids 18 

A  Sphere 18 

Solids  with  Straight  Lines , 18 

Solids  with  Curved  Lines 18 

Solids  with  both  Straight  and  Curved  Lines 21 

Screw-threads. 21 

Conventional  and  Exact  Methods  of  Representation 21 

10.  Dimensioning 24 

11.  Remarks x 27 

CAVALIER   PROJECTION. 

12.  Introductory 30 

13.  Theory 30 

14.  Application  of  the  Theory 30 

15.  M*ethod  of  Procedure 33 

16.  Flexibility 33 

1 7.  Practical  Examples 34 

Solids  with  Straight  and  Curved  Lines 34 

Screw-threads 36 

18.  Distortion • 37 


CHAPTER  II. 

SHADOWS. 

19.  Intrqductory 38 

20.  Theory : 3$ 

21.  The  Shadow  of  a  Point 40 

22.  The  Shadow  of  a  Right  Line 41 

The  Shadow  on  One  Plane  only 42 

The  Shadow  on  both  Planes 42 

The  Shadow  of  Lines  Parallel  to  V  and  H 43 

23.  The  Shadow  of  a  Curved  Line 44 

24.  The  Shadow  of  Solids 45 

Plane  Surfaces 45 

The  Shadow  on  the  Planes-  only 45 

The  Shadow  on  both  the  Object  and  the  Planes 47 

Single  Curved  Surfaces 49 

The  Shadow  of  Straight  and  Curved  Lines  on  the  Surface 49 

The  Shadow  of  the  Surface  on  the  Planes 49 

Double  Curved  Surfaces 51 

The  Shadow  on  the  Surface  and  on  the  Planes 51 

25.  Remarks.  .  . 55 


CONTENTS.  vii 

CHAPTER  III. 

PERSPECTIVE. 
SECTION  PAGE 

26.  Definition 56 

27.  Perspective  and  Mechanical  Drawing  Compared 56 

28.  Mechanical  and  Free-hand  Perspective ^6 

29.  Perspective  as  Applied  by  the  Engineer 57 

30.  Theory  of  Perspective 57 

31.  The  Perspective  of  a  Point 59 

32.  The  Perspective  of  a  Right  Line 61 

33.  The  Perspective  of  a  Curved  Line 61 

34.  Why  Objects  are  Assumed  in  the  Second  Quadrant 61 

35.  The  Perspective  of  an  Indefinite  Right  Line 64 

36.  The  Vanishing-point  of  a  "Line 66 

37.  Rule  for  Finding  the  Vanishing-point  of  a  Line 66 

38.  Rule  for  Finding  the  Perspective  of  a  Line 68 

39.  The  Diagonal  and  Perpendicular 68 

40.  Conventional  Method  for  Finding  Perspectives 70 

41.  The  Horizon-line 73 

42.  Distance  Points 7- 

43.  The  Perspective  of  a  Plane  Figure 76 

44.  Special  Cases  of  the  Right  Line 76 

45.  The  Plan  and  Elevation  Removed  from  the  Field  of  the  Picture 78 

46.  Practical  Perspective 81 

47.  Parallel  Perspective 85 

48.  Oblique  or  Angular  Perspective 85 

49.  How  to  Assume  Conditions 88 

50.  The  Perspective  of  Shadows oo 

Theory 9I 

Application pr 

Examples 03 

Remarks O5 


PART  II. 

EXERCISES. 


CHAPTER  IV. 

THEORETICAL  PROBLEMS. 

51.  Explanatorv .......    ....     07 

52.  General  Directions 97 

POINTS,  LINES,  AND  PLANES. 

53.  Problem  i.  The  Assumption  of  Points  and  Lines 09 

54.  Problem  2.  The  Assumption  of  Lines  Continued,  and  the  Assumption  of 


viii  CONTENTS. 

SBCTION  PAGE 

55.  Problem  3.  The  Revolution  of  a  Point,  the  Finding  of  the  Traces  of  a 

Plane,  and  the  Intersection  of  Planes.  ...    102 

56.  Problem  4.  The  Passage  of  Planes,  and  the  Finding  of  True  Lengths. ...  103 

57.  Problem  5.  The  Projection  of  Lines  on  an  Oblique  Plane 104 

58.  Problem  6.  The  Angle  between  a  Line  and  a  Plane 105 

59.  Problem  7.  The  Angle  between  Two  Planes 106 

60.  Problem  8.  The  True  Distance  between  Two  Lines 107 

61.  Problem  9.  To  draw  a  Circle  through  Three  Points 109 

TANGENT    PLANES. 

62.  Problem  10.  Tangent  Planes  to  Cones  and  Cylinders no 

63.  Problem  n.  Tangent  Planes  to  Cones  and  Cylinders,  Continued no 

64.  Problem  12.  A  Tangent  Plane  to  a  Sphere in 

65.  Problem  13.  A  Tangent  Plane  to  a  Warped  Surface 112 

66.  Problem  14.  A  Plane  through  a  Line  and  Tangent  to  a  Sphere 113 

67.  Problem  15.  A  Tangent  Plane  to  a  Convolute  Surface 113 

68.  Problem  1 6.  A  Tangent  Plane  to  a  Double  Curved  Surface  of  Revolution .  115 

INTERSECTIONS. 

69.  Problem  17.  The  Intersection  of  a  Cone  and  a  Plane,  and  of  a  Cone  and 

Cylinder 116 

70.  Problem  1 8.  The  Intersection  of  a  Cylinder  and  Plane,  and  of  Two  Cones .    116 

71.  Problem  19.  The  Intersection  of  a  Sphere  and  a  Plane,  and  of  Two 

Cylinders 117 

DEVELOPMENTS . 

72.  Problem  20.  The  Development  of  an  Oblique  Cone 118 

73    Problem  21.  The  Development  of  an  Oblique  Cylinder 118 

74.  Problem  22.  The  Development  of  a  Right  Cylinder 119 

75.  Problem  23.  The  Development  of  a  Convolute  Surface 120 

76.  Problem  24.  The  Development  of  a  Sphere 120 

CHAPTER  V. 

PRACTICAL  PROBLEMS. 

77.  Explanatory 121 

78.  General  Directions 121 

TRUE   LENGTHS,    TRUE   ANGLES,    INTERSECTIONS,    DEVELOPMENTS,    ETC. 

79.  Problem  i.  To  Lay  Out  the  Cutting  Lines  for  getting  out  the  Wreath, 

Starting  from  a  Newel-post ' 122 

80.  Problem  2.  To  Show  the  Layout  for  the  Shop  for  a  Wrought-iron  Support  124 

8 1 .  Problem  3.  To  Locate,  and  to  Find  the  Length  of  Guy- wires  for  a  Smoke- 

stack     126 

82.  Problem  4.  To  Find  the  Shape,   Size,  and  Bevels  for  an  Example  in 

Cabinet  Work.  .  128 


CONTENTS.  ix 

SECTION  PAGB 

83-85.  Problems,  5,  6,  and  7.  To  Lay  Out  Certain  Sheets  of  a  Locomotive- 
boiler 132,  134,  137 

86.  Problem  8.  To  Find  the  Shape,  and  Size  of  the  Plates  used  to  Form  an 

Elbow .138 

87.  Problem  9.  To  Lay  Out  the  Sheets  for  a  Reducing  Breeching 140 

88.  Problem  10.  To  Lay  Out  the  Plates  for  a  Screw-grain  Conveyor 144 

89.  Problem  n.  To  Find  the  Shape  and  Size  of  Certain  Plates  Forming  Part 

of  a  Positive-feed  Mechanism .146 

90.  Problem  12.  To  Find  the  Angle  for  a  Special  Angle-iron  for  Framing  the 

Corners  of  a  Coal-hopper 148 

SHADOWS. 

91-93.  Problems  13,  14,  and  15.  To  Find  some  Elementary  Shadows.. 150 

94.  Problem  16.  To  Find  the  Shadow  Cast  by  a  Taboret 154 

95-96.  Problems  17  and  18.  To  Find  the  Shadow  on  a  Double  Curved  Surface  154 

PERSPECTIVE. 

97.  Problem  19.  To  Find  some  Elementary  Perspectives 156 

98.  Problem  20.  To  Find  the  Perspective  of  a  Flight  of  Stone  Steps 158 

99.  Problem  21.  To  Find  the  Perspective  of  a  Small  Railway  Station-house. .  160 
TOO.  Problem  22.  To  Find  the  Perspective  of  a  Railway  Arch 163 

101.  Problem  23.  To  Find  the  Perspective  of  an  Architectural  Arch 164 

102.  Problem  24.  To  Find  the  Perspective  of  a  Small  Dwelling-house 167 

103.  Problem  25.  To  Find  the  Isometric  Perspective  of  a  House 1 72 

SUPPLEMENTAL. 
•  EXERCISES  IN  LETTERING. 

104.  Explanatory 174 

A  Sheet  of  Free-hand  Letters 175 

A  Shop  Card 176 

A  Cover  Sheet 177 


ADVANCED     MECHANICAL     DRAWING. 


PART  I. 

THEORY,  DEFINITIONS,  ETC. 
CHAPTER  I. 

ISOMETRIC    DRAWING. 

i.  Definition. — Isometric  drawing  is  that  branch  of  mechani- 
cal drawing  which  enables  one  to  represent  an  object  in  such  a 
manner  as  to  present  three  sides  or  faces  in  a  single  drawing  or 
view.  That  is,  such  a  drawing  serves  the  same  end  as  an  ordinary 
three-view  mechanical  drawing;  furthermore,  it  pictures,  after  a 
fashion,  the  object  as  it  would  appear  if  placed  before  the 
observer,  and  because  of  this  characteristic  is  legible  to  one  not 
able  to  read  a  mechanical  drawing. 

Isometric  drawing  is  also  called  "isometric  perspective" 
and  " practical  perspective";  it  is  called  isometric  perspective 
because  it  pictures  an  object  as  a  whole,  and  practical  perspective 
because  of  its  greater  simplicity  as  compared  with  perspective 
drawing. 

It  is  really  a  joint  between  ordinary  mechanical  drawing 
and  perspective  drawing,  since  it  contains  features  of  each.  For 
example,  lines  which  are  drawn  parallel  in  the  mechanical  draw- 
ing of  an  object  are  also  drawn  parallel  in  the  isometric  drawing 


2  ADVANCED  MECHANICAL   DRAWING. 

of  the  object,  and  since  the  isometric  drawing  shows  three  faces 
it  is  a  pseudo  perspective. 

2.  Theory. — If  a  cube  be  held  in  a  position  such  that  one  of 
its  diagonals  is  perpendicular  to  one  of  the  planes  of  projection 
(Fig.  i),  its  projection  on  that  plane  is  said  to  be  an  isometric 
projection  (since  all  the  lines  of  the  cube  are  equally  foreshortened 
in  the  projection  because  of  their  uniform  inclination  to  the 
plane) — the  term  "isometric"  meaning  "in  equal  parts." 


FIG.  i. 


The  figure  shows  the  projection  on  the  vertical  plane  (either 
plane  of  projection  may  be  used)  and  shows  three  visible  faces. 
This  is  sufficient  for  all  practical  purposes,  and  it  is  customary 
to  disregard  the  other  projection  entirely  and  thus  eliminate 
all  reference  to  the  planes  of  projection. 

Now  any  object  may  be  considered  as  inclosed  within  a  cube 
(a  side  of  the  cube  would  equal  the  greatest  dimension  of  the 
object)  and,  thus  considered,  it  can  be  projected  with  the  cube. 


ISOMETRIC  DRAWING.  3 

3.  Explanation  of  Terms. — In  Fig.  2,  which  is  a  mechanical 
drawing  of  the  arrangement  shown  in  Fig.  i,  note  the  central 
point  of  the  front  elevation:  this  point  is  called  the  origin;  the 
three  full  lines  radiating  from  it,  the  isometric  axes;  the  planes 


FIG.  2. 

determined  by  the  isometric  axis  are  called  isometric  planes,  and 
all  lines  in  these  planes  drawn  parallel  to  the  isometric  axes  are 
called  isometric  lines. 

The  isometric  axes — three  lines  120°  apart  (Fig.  3) — form 
the  basis  for  all  isometric  drawing. 

4.  Distinction  between  Isometric  Projection  and  Isometric 
Drawing. — The  lines  of  the  projection  of  the  cube  in  Figs,  i  and 
2  are  about  .8  of  the  length  of  the  original;  that  is,  the  size  of 


4  ADVANCED  MECHANICAL  DRAWING. 

the  projection  is  about  .8  of  the  size  of  the  cube.  If  the  projection 
is  drawn  the  same  size  as  the  cube,  it  will  represent  the  projection 
of  a  cube  about  1.25  times  the  size  of  the  original.  Since  it  would 
require  special  scales  to  obtain  .8  of  the  several  dimensions  of 
an  object,  it  is  common  practice  to  construct  the  drawing  to  rep- 
resent the  projection  of  an  object  1.25  larger  than  the  original; 
that  is,  dimensions  are  taken  as  in  ordinary  mechanical  drawing 
— full  size,  one-half  size,  one-quarter  size,  etc.  The  drawing 
so  constructed  is  called  an  isometric  drawing  to  distinguish  it 


FIG.  3. 

from  the  isometric  projection,  the  dimensions  of  which  would 
be  .8  of  those  of  the  isometric  drawing;  in  short,  an  isometric 
drawing  is  one-fourth  larger  than  an  isometric  projection. 

5.  Isometric  Scales. — The  uniform  angle  made  by  the  lines 
of  the  cube  with  the  plane  of  projection  in  Fig.  i  is  an  angle 
of  35°  16';  now,  if  a  scale  with  full-size  divisions — one  inch  equal 
to  one  inch — be  placed  at  such  an  angle  with  another  scale,  as 
is  shown  by  Fig.  4,  the  divisions  on  the  inclined  scale  can  be 
projected  onto  the  upright  scale  the  same  as  the  lines  of  the  cube 
are  projected  onto  the  plane  of  projection,  and  a  scale  of  fore- 
shortened divisions  obtained — an  isometric  scale.  In  the  figure 
the  left  scale  of  B  is  projected  onto  the  right  side  of  scale  A. 

On  the  other  hand,  if  full-size  divisions  be  projected  from 


ISOMETRIC  DRAWING.  5 

the  upright  scale  onto  the  inclined  scale,  as  is  shown  in  the  figure 
by  the  projections  from  the  left  side  of  scale  A  to  the  right 
side  of  Bf  a  series  of  divisions  will  be  obtained  which  are  1.25 
larger — a  second  form  of  isometric  scale. 

Such  scales  as  these  would  be  used  to  construct  isometric 
projections;  using  scale  A,  the  object  would  be  measured  with 
the  scale  on  the  left  and  the  projection  constructed  from  the 
scale  on  the  right;  or,  using  scale  B,  the  object  would  be  measured 


FIG.  4. 


with  the  scale  on  the  right  and  the  projection  constructed  from 
the  scale  on  the  left. 

As  already  stated  these  scales  are  special  and  are  not  used 
in  ordinary  practice,  the  usual  architect's  or  engineer's  scale 
being  used,  and  the  projection  made  full  size  or  to  scale,  as  already 
described — an  isometric  drawing. 

6.  Practical  Application  of  the  Theory.— To  consider  an 
object  as  inclosed  within  a  cube,  a  side  of  which  would  equal 
the  greatest  dimension  of  the  object,  would  involve  the  draught- 
ing of  a  number  of  lines  in  the  execution  of  the  isometric  repre- 
sentation which  would  be  longer  than  necessary  and  require 
greater  space  for  the  drawing.  In  practice  the  inclosing  figure 
is  a  rectangular  "box,"  the  three  dimensions  of  which  conform 


6  ADVANCED  MECHANICAL  DRAWING. 

with  the  three  dimensions — length,  breadth,  and  thickness — 
— of  the  object;  the  application  of  the  theory  is  seen  by  taking 
a  side  of  the  inclosing  box  and  on  it  constructing  a  cube,  then 
arranging  the  cube  with  the  correct  reference  to  a  plane  of 
projection  and  projecting  the  cube  and  the  rectangular  box 
together  with  its  contents,  as  witness  Fig.  5:  the  figure  0-1-2-3-4- 


2i 


FIG.  5. 

5-6-7  representing  the  cube,  the  corners  of  which  are  numbered 
the  same  as  the  corners  of  the  cube  in  Fig.  i. 

7.  Method  of  Procedure. — The  above  is  carried  out  in  practice 
as  follows: 

Let  it  be  required  to  construct  an  isometric  drawing  of  the 
object,  the  mechanical  drawings  for  which  are  shown  in  A,  Fig. 
6.  The  first  step  is  to  draw  the  isometric  axes — three  lines  120° 
apart,  as  shown,  and  so  taken  (one  line  perpendicular  and  the 
other  two  at  30°  with  the  horizontal)  for  convenience  in  draw- 


ISOMETRIC  DRAWING. 


ADVANCED  MECHANICAL  DRAWING. 


ing — and  select  (arbitrarily)  one  axis  to  represent  length,  one 
for  breadth,  and  one  for  thickness  or  height — /,  b,  and  A;  the 
second  step  is  to  ascertain  the  three  dimensions  of  the  inclosing 
box  for  the  object  by  inclosing  the  mechanical  drawings  within 
rectangles,  then  lay  off  these  dimensions  on  the  isometric  axes, 
and  draw  the  isometric  drawing  of  the  inclosing  box;  the  third 
step  is  to  select  one  face  of  the  box  and  draw  in  it  all  the  lines 
of  the  inclosed  object  showing  there;  the  fourth  step  is  to  simi- 
larly treat  a  second  face;  and  so  on,  until  all  of  the  faces  have 
been  treated,  and  all  of  the  lines  of  the  object  drawn;  the  last 
step  is  to  erase  all  construction  lines. 

8.  Flexibility. — The  assumption  of  the  isometric  axes  is  the 
first  step  in  all  isometric  drawing;   they  may  be  assumed  in  any 

ISOMETRIC  AXES 
Note  the  several  posi 

tions,B,C,D,E,F. 
1  i 


position  so  long  as  their  proper  relation,  one  to  another,  is  main- 
tained. It  is  obvious,  of  course,  that  for  practicability  a  position 
must  be  assumed  to  suit  the  drawing  instruments,  that  is,  one 
;axis  should  always  be  upright,  horizontal,  at  30°  or  60°  with  the 


ISOMETRIC  DRAWING.  9 

horizontal,  etc.,  and  the  other  two  axes  drawn  at  120°  with  it. 

In  practice  the  position  of  the  axes  is  determined,  in  a  measure 
by  the  object  itself;  that  is,  the  isometric  axes  determine  the 
isometric  planes,  and  the  planes  are  determined  by  the  faces  of 
the  object  which  are  to  be  illustrated. 

As  evidence  of  the  above,  assume  that  an  object  is  to  be  drawn 
which  can  be  inclosed  in  a  rectangular  box,  and  note  the  several 
positions  which  the  box  may  assume,  as  shown  in  Fig.  7;  also 
note  the  other  isometric  drawings  of  the  text. 

9.  Practical  Examples. — As  a  further  exposition  of  the  subject, 
it  is  proposed  to  discuss  the  execution  of  a  number  of  representa- 
tive examples  in  drawing. 

Plane  figures. — A  square.  Let  it  be  required  to  represent 
a  2"  square,  Fig.  8,  in  "  isometric,"  and,  furthermore,  let  it  be  re- 
quired to  represent  it  with  its  plane  horizontal. 

Now,  in  isometric  drawing  a  horizontal  plane  is  determined 
by  lines  which  are  at  30°  with  the  horizontal;  hence  to  draw  the 
square,  draw  the  two  lines  A-B  and  A-D,  as  shown,  each  making, 
an  angle  of  30°  with  the  horizontal,  and  from  their  point  of  inter- 
section lay  off  on  each  line  a  length  equal  to  2" — the  length  of 
a  side  of  the  square;  through  the  point  thus  determined  on  A -B, 
point  B,  draw  a  line  B-C  parallel  to  the  line  A-D,  and  through 
the  point  D  of  A-D,  similarly  determined,  draw  a  line  D-C  paral- 
lel to  A-B  to  an  intersection  with  B-C.  The  line  B-C  is,  clearly, 
equal  to  A-D,  which  is  equal  to  a  side  of  the  square,  being  paral- 
lel lines  comprehended  between  parallels;  also,  for  the  same 
reason,  the  line  D-C  is  equaL4-#,  a  second  side  of  the  square; 
therefore,  the  figure  A-B-C-D  is  the  required  isometric. 

The  term  " horizontal  plane"  as  applied  to  the  plane  deter- 
mined by  lines  at  30°  with  the  horizontal  is  an  arbitrary  one,  and 
is  used  to  distinguish  it  from  planes  determined  by  a  vertical 
line  and  a  line  at  30°  with  the  horizontal,  which  for  the  purpose 
of  discussion  will  be  termed  "vertical  planes."  Both  terms, 
however,  are  misnomers,  as  the  planes  are  neither  horizontal 
nor  vertical,  but  are  oblique  planes,  as  witness  Fig.  i :  the  hori- 
zontal plane  corresponds  to  the  plane  of  the  top  or  bottom  of 


10 


ADVANCED  MECHANICAL   DRAWING. 


the  cube  (0-1-2-3  or  4-5-6-7),  and  the  vertical  plane  to  the  side 
faces  of  the  cube  (0-3-7-4,  etc).  In  like  manner,  a  plane  which 
is  determined  by  a  horizontal  line  and  a  60°  line  or  by  two  60° 
lines  will  be  termed  an  " oblique  plane."  (See  E  and  F,  Fig.  7.) 

A  circle.  Let  it  be  now  required  to  draw  a  circle  on  a  hori- 
zontal plane.  This  is  accomplished  by  first  assuming  the  circle 
to  be  inclosed  within  a  square,  then  drawing  the  square  in  iso- 
metric, and  proceeding  as  follows: 

If  the  circle  is  inclosed  within  a  square,  it  is  obvious  that  it  will 
be  tangent  to  each  side  of  the  square  at  its  middle  point ;  therefore, 


locate  these  points  by  drawing  the  center  lines  of  the  square,* 
as  shown  in  Fig.  8,  then  with  a  corner  of  the  square  as  a  center 
and  a  radius  equal  to  the  distance  from  this  point  to  the  middle 
point  of  an  opposite  side  of  the  square,  draw  a  circular  arc  be- 
tween the  two  opposite  middle  points;  next  reverse  the  operation, 
that  is,  use  the  opposite  corner  of  the  square  as  a  center,  etc.; 
these  two  arcs  drawn,  draw  the  diagonal  of  the  square  and  draw 
the  lines  determining  the  above  radii:  the  points  where  these 
lines  cut  the  diagonal  will  be  new  centers,  and  with  new  radii  rep- 
resented by  the  distance  from  one  of  these  points  to  the  nearest 
middle  point  of  a  side  of  the  square,  other  circular  arcs  may 


*  The  isometric  drawing  of  the  square  is  not  a  square,  of  course;  the  terms 
square,  rectangle,  etc.,  are  used,  however,  the  same  as  in  ordinary  mechanical 
drawing. 


ISOMETRIC  DRAWING. 


II 


be  drawn  and  a  closed  curve  obtained  representing  the  circle  in 
isometric.     The  construction  is  clearly  shown  by  the  figure. 

An  isometric  view  being  an  oblique  view,  it  is  obvious  that 
the  representation  is  an  ellipse.  The  above  is  not  a  true  rep- 
resentation, but  is  an  approximation,  and  is  the  usual  practice 
in  isometric  because  of  its  ease  of  execution.  The  figure  shows 
its  application  in  the  horizontal  plane;  it  is  applied  in  exactly  the 
same  manner  in  all  isometric  planes.  (See  Fig.  9.) 


FIG.  9. 


A  triangle.  Fig.  10  shows  a  triangle,  A  being  the  mechanical 
drawing  and  B  the  isometric  drawing.  To  construct  the  figure 
inclose  the  mechanical  drawing  within  a  rectangle,  then  draw 
the  rectangle  in  isometric;  now  the  figure  shows  the  triangle 
inclosed  in  such  a  manner  that  one  of  its  sides  forms  a  side  of 
the  inclosing  rectangle,  and  the  triangle  is  such  that  the  apex 
of  the  angle  opposite  this  side  is  at  the  middle  point  of  the  op- 
posite side  of  the  rectangle;  hence,  to  draw  the  triangle  within 


12 


ADVANCED  MECHANICAL  DRAWING. 


the  isometric  rectangle,  draw  the  coincident  side  and  join  the 
extremities  with  the  middle  point  of  the  opposite  side  of  the 
rectangle. 


FIG.  io. 


It  will  be  remarked  that  the  figure  is  drawn  on  a  horizontal 
plane;  the  construction  is  the  same  for  any  plane. 


FIG.  ii. 


A  hexagon.    In  Fig.  n,  A  is  a  mechanical   drawing   of  a 
hexagon;   to  draw  it  in  isometric,  inclose  the  hexagon  within  a 


ISOMETRIC  DRAWING-  13 

rectangle,  then  draw  the  rectangle  as  shown  in  B\  now  if  the 
diameters  of  the  hexagon  be  drawn,  the  extremes  of  the  long 
diameter — the  " diagonal"  of  the  hexagon — will  be  two  points 
in  the  hexagon,  and  the  extremes  of  the  short  diameter — the 
distance  between  "flats" — will  be  the  middle  points  of  two 
opposite  sides  of  the  hexagon;  these  two  sides  are  coincident 
with  two  of  the  sides  of  the  inclosing  rectangle,  and  by  laying 
off  from  these  middle  points  distances  on  the  sides  equal  to  one- 
half  of  the  length  of  a  side,  as  shown,  the  remaining  four  corners 
of  the  hexagon  are  obtained;  the  hexagon  is  then  drawn  by 
connecting  the  six  points. 

The  isometric  plane  in  this  example  is  a  vertical  plane,  and 
corresponds  to  the  0-3-7-4  plane  of  the  cube  in  Fig.  i. 

An  octagon.  A  method  for  drawing  an  octagon  in  isometric 
is  shown  by  Fig.  12.  The  inclosing  rectangle  is  drawn  as  in 


FIG.  12. 

the  other  examples,  and  the  four  points  of  the  figure  which  touch 
the  sides  of  the  rectangle  are  found  by  drawing  the  center  lines 
as  shown;  the  remaining  four  points  are  found  by  projecting 
(horizontally  and  vertically — parallel  with  the  sides  of  the  inclos- 
ing rectangle)  the  points  of  the  mechanical  drawing  onto  the 
sides  of  the  inclosing  rectangle,  then  transferring  these  divisions 
to  the  sides  of  the  isometric  rectangle,  and  through  them  draw- 


14  ADVANCED  MECHANICAL  DRAWING. 

ing  the  isometric  lines  as  shown.  The  intersections  will  define 
the  location  of  the  points.  The  octagon  is  then  drawn  by  con- 
necting the  eight  points  as  shown. 

This  method  of  locating  points  by  intersecting  lines  cor- 
responds to  the  method  of  locating  points  by  ordinate  and 
abscissa  of  analytic  geometry,  and  is  much  used  in  isometric 
construction;  for  the  purpose  of  discussion  it  will  be  referred 
to  as  plotting. 

The  plane  of  this  figure  is  a  vertical  plane  and  corresponds 
to  the  plane  0-1-5-4  of  Fig.  i. 

A  star.  Fig.  13  shows  a  method  of  plotting  applied  in  laying 
out  a  star. 


FIG.  13. 

An  ellipse.     Fig.  14  shows  the  layout  of  an  ellipse. 

A  parabola.     Fig.  15  illustrates  a  parabola. 

Any  irregular  figure.     Fig.  16. 

A  special  irregular  figure. — Circular  arcs.  Fig.  17  illus- 
trates a  method  for  use  when  the  figure  is  made  up  of  circular 
arcs.  The  inclosing  rectangle  is  obtained  and  drawn  in  the 


ISOMETRIC  DRAWING. 


FIG.  14 


FIG.  15 


i6 


ADVANCED  MECHANICAL   DRAWING. 


usual  manner;  in  this  case  the  plane  is  an  oblique  plane  and 
corresponds  to  the  plane  0-1-2-3  of  E,  Fig.  7.  The  semicircle 
at  the  left  end  is  found  by  laying  out  an  inclosing  rectangle 


FIG.  i 6. 


A-H-P-D,  each  side  of  which  is  drawn  i  J"  long,  and  the  semi- 
circle E-U-E  drawn  by  means  of  two  circular  arcs,  found  as  in 
the  approximate  method  given  for  drawing  a  circle  (page  10). 


FIG.  17. 

The  f"  arc  is  obtained  by  drawing  the  inclosing  rectangle  F-I~ 
N-Oy  each  side  of  which  is  J"  long,  from  which  the  radius  is 
found  and  the  arc  drawn  as  shown.  The  other  arcs  are  drawn 
in  a  similar  manner,  as  illustrated  by  the  drawing. 


ISOMETRIC  DRAWING. 


From  the  above  the  following  rule  is  deduced:  To  draw 
any  part  oj  a  circle,  consider  it  as  a  whole  circle;  draw  the  inclosing 
rectangle  and  find  the  centers  and  radii  as  if  going  to  draw  the 
entire  circumference,  then  use  only  the  center  and  radius  necessary 
lor  the  desired  portion. 

PLAN 


ELEVATION 


FIG.  18. 


Curves. — A  helix.  A  helix  or  any  curve  which  is  not  a 
plane  curve  may  be  drawn  as  follows : 

Plot  the  curve  as  shown  by  the  mechanical  drawing  A  of 
Fig.  18  and  obtain  the  dimensions  of  the  inclosing  box;  next, 
draw  the  box  in  isometric,  then  plot  the  plan  of  the  curve  in  the 
proper  isometric  plane,  as  is  indicated  by  the  numbered  dots; 
this  done,  draw  lines  parallel  to  the  remaining  isometric  axis 
(two  have  been  used  to  define  the  plane  of  the  plan)  through 


i8 


ADVANCED  MECHANICAL   DRAWING. 


each  point  or  dot,  and  on  these  lines  lay  off  lengths  to  cor- 
respond to  the  plotting  of  the  mechanical  drawing.  The  points 
thus  denned  will  be  the  locus  of  the  required  curve. 

Solids. — A  sphere.  A  sphere  is  drawn  in  isometric  as  a 
true  circle,  the  radius  of  which  is  equal  to  one-half  of  the  major 
axis  of  the  isometric  representation  of  a  circle  of  the  same  diame- 
ter, as  witness  Fig.  19.  The  figure  shows  the  layout  for  three 
great  circles  of  a  sphere;  now  it  is  evident  that  the  isometric 


FIG.  19. 

drawing  of  the  sphere  must  contain  all  of  its  great  circles,  hence 
the  center  and  radius  as  shown. 

A  solid  with  straight  lines  only.  When  the  b'nes  of  an 
object  are  all  straight  lines,  the  execution  of  the  drawing  is  very 
simple,  and  particularly  so  when  most  of  the  lines  are  parallel 
with  the  isometric  axes,  as  is  the  case  in  the  dovetail  shown  in 
Fig.  20.  To  draw  the  pieces  forming  the  joint,  draw  the  inclosing 
boxes  and  in  these  lay  out  the  lines  of  the  figure  as  shown. 

Solids  with  curved  lines.  Fig.  21  illustrates  the  layout  for 
a  small  collar.  The  inclosing  box  is  obtained  in  the  usual  man- 
ner, and  the  drawing  constructed  in  it  as  follows : 

Draw  the  center  lines  of  one  end  face,  and  on  these  construct 


ISOMETRIC  DRAWING. 


5 1 

"«|oo 


FIG.  20. 


20 


ADVANCED  MECHANICAL  DRAWING. 


the  inclosing  rectangles  of  the  circles  as  shown;  the  ellipses 
are  then  drawn  as  described  on  page  10  and  as  indicated  by 
the  drawing.  The  other  end  face  may  be  laid  out  in  a  similar 
manner,  or  by  a  shorter  method  as  follows:  Having  the  centers 
of  the  arcs  for  the  ellipse  in  one  end  face,  and  knowing  the  dis- 
tance (2")  between  the  planes  of  the  end  faces,  move  the  centers 
in  the  proper  direction,  and  parallel  with  the  axis  of  the  collar, 


FIG.  21. 

a  distance  equal  to  this  dimension,  then  with  the  new  centers 
and  the  same  radii  as  used  for  the  other  end  describe  the  neces- 
sary arcs.  The  ends  drawn,  the  figure  is  completed  by  drawing 
tangent  lines  as  shown.  The  figure  shows  the  construction 
clearly,  also  a  method  of  showing  a  half-section  in  isometric. 

Fig.  22,  depicting  all  of  the  lines  of  construction,  illustrates 
a  method  of  executing  a  figure  when  the  circles  are  in  different 
planes.  The  characteristic  of  the  method  is  the  use  of  a  longi- 
tudinal center  line  on  which  the  elevation  or  position  of  the  several 
planes  is  made  manifest  by  the  location  of  their  center  points; 


ISOMETRIC  DRAWING. 


21 


these  points  determined,  the  center  lines  of  the  rectangles,  the 

rectangles,  and  ellipses   are   drawn  substantially  as  already  de- 
scribed, and  as  shown  in  the  drawing. 

An    object    with    both    straight    and   curved  lines.     Fig.   23 


FIG.  22. 

shows  a  method  of  representing  an  object  involving  the  drawing 
of  both  straight  and  curved  lines;  the  construction  is  obvious 
from  the  figure. 

The  drawing  is  a  conventional  representation  of  Fig.  5,  and 
also  a  further  exposition  of  section  2,  the  cube  A-B-C-D-E-F-G-H 
being  laid  out  on  the  2tt"  dimension  of  the  inclosing  box.  The 
extra  lines  shown  are  not  necessary,  of  course,  but  are  given 
for  the  above  purpose. 

The  representation  of  screw-threads.  The  true  representation 
of  screw-threads  is  quite  complicated,  and  is  rarely  done;  the 


22 


ADVANCED  MECHANICAL   DRAWING. 


usual  practice  is  the  convention  illustrated  by  Fig.  24.  The 
threads  in  the  nut  are  drawn  by  first  constructing  an  ellipse  in 
the  top  plane  representing  a  circle  of  the  diameter  of  the  bolt 
at  the  root  of  the  thread,  then  projecting  the  centers  used  in  the 
direction  of  the  axis  of  the  nut,  and  drawing  a  series  of  parallel 
circular  arcs  as  shown.  For  example,  if  the  thread  be  an  eight- 
pitch  thread — eight  threads  to  one  inch — lay  off  a  series  of  divisions 


A 

j 

Nt 

T 

j 

t 

/" 

3 

X 

j 

-    ; 

^ 

If 

<i> 

2|"- 



-^ 

PL 

AN. 

V 

?r;; 

T 

$ 

j 

1* 

ys 

1 

| 

i 

FIG.  23. 

on  the  line  of  centers  which  are  A"  apart;  the  points  thus  obtained 
will  be  the  necessary  center  points.  It  is  obvious  that  the  distance 
between  centers  will  be  one-half  of  the  distance  between  the 
points  of  the  threads,  there  being  a  center  necessary  for  both 
the  top  and  bottom  of  the  thread.  (Understand,  this  is  not  a 
true  representation,  but  a  convenient  convention.) 

The  thread  on  the  bolt  is  drawn  in  a  similar  manner. 

Fig.  25  illustrates  a  true  representation  of  a  thread,  the  curve 
(a  helix)  of  both  top  and  bottom  of  the  "V"  having  been  plotted. 

It  is  well  to  note  the  layout  of  the  nut  in  this  example,  as 
the  construction  is  a  typical  one.  The  inclosing  solid  is  drawn 


ISOMETRIC  DRAWING.  23 

AS  usual,  and  the  hexagon  of  the  base  laid  out  as  described  on 
page  12;  this  done,  at  each  corner  of  the  hexagon  erect  a 
line  perpendicular  to  the  plane  of  the  base  and  on  it  lay  off  a 


FIG.  24. 

length  equal  to  an  edge  of  the  nut;  next,  bisect  each  side  of  the 
hexagon  and  at  the  points  of  bisection  erect  other  perpendiculars 
and  on  them  lay  off  lengths  equal  to  the  length  of  the  face  of  the 
nut  at  this  point;  this  will  give  three  points  of  the  curved  edge 


24  ADVANCED  MECHANICAL   DRAWING. 

of  each  face,  and  through  these  ,a  curve  may  be  drawn  with  the 
irregular  curve ;  the  layout  in  the  top  plane  is  obvious. 


FIG.  26. 


Fig.  26  illustrates  a  single,  square  thread. 
10.  Dimensioning. — In  ordinary  orthographic  projection  the 
drawings    are    dimensioned  in    two    directions,  horizontally    and 


ISOMETRIC  DRAWING. 


25 


vertically;   in  isometric  drawing  the  drawings  are  dimensioned 
in  three  directions — parallel  with  the  three  isometric  axes. 


FIG.  27. 

An  isometric  drawing  intended  for  shop  purposes  should  be 
well  and  completely  dimensioned,  as  such  a  drawing  is  difficult 


26 


ADVANCED  MECHANICAL   DRAWING. 


to  scale;   if  it  is  scaled,  however,  the  scaling  should  be  done  in 
directions  parallel  with  the  axes. 

The  planning  of  the  dimensioning  is  a  matter  of  some  moment, 
as  the  legibility  of  the  drawing  is  dependent  directly  upon  it. 


FIG.  28. 

Where  possible  the  dimension  lines  and  figures  should  be  so 
arranged  as  to  appear  to  lie  flat  on  the  plane  containing  the 
representation  of  the  part  .dimensioned.  A  number  of  the  pre- 
ceding figures  are  dimensioned  to  illustrate  this  point,  also,  that 
they  may  serve  as  a  copy  for  practice  in  acquiring  the  art;  it  is 


FIG.  29. 


28  ADVANCED  MECHANICAL  DRAWING. 

not  always  possible  or  convenient,  however,  to  place  the  dimen- 
sions in  the  proper  plane,  and  Figs.  27  and  28  are  given  as  ex- 
amples to  be  followed  in  such  cases.  Fig.  29  is  given  as  an 
example  for  the  student  to  dimension. 

11.  Remarks. — In  mechanical  drawing  one  finds  coordinate 
ruled  paper  a  great  convenience  for  sketch-work;  it  may  be  used, 
also,  for  isometric  sketches,  as  witness  Fig.  30,  though  for  this 
purpose  a  specially  ruled  paper  (sometimes  called  "Iso"  paper) 
is  to  be  had  of  the  trade.     (See  Fig.  31.) 

In  viewing  Figs.  23  and  26  the  eye  seems  to  recognize  that 
receding  lines  should  converge,  and  the  drawing  appears  distorted. 
This  feature  of  isometric  drawing — distortion  because  of  paral- 
lel lines — is  one  of  the  objectionable  features  of  the  art,  and  one 
should  exercise  his  judgment  in  its  use,  using  some  other  method 
of  representation  in  cases  where  the  object  is  of  such  character 
as  to  produce  marked  distortion. 

The  examples  given  have  necessarily  referred  directly  to  the 
mechanical  drawing  of  the  object  under  consideration;  in  practice 
it  is  often  necessary  to  first  execute  a  mechanical  drawing  of  the 
object  before  the  isometric  drawing  can  be  drawn;  there  are,  how- 
ever, many  cases  where  the  drawing  can  be  executed  directly  from 
the  object.  In  such  cases  the  use  of  an  inclosing  box  is  dispensed 
with  and  the  drawing  is  laid  out  with  reference  to  a  center  or 
base  line  as  in  ordinary  drawing,  the  dimensions,  of  course,  being 
taken  parallel  with  the  isometric  axes.  It  is  well,  however,  to 
always  use  the  inclosing  box,  as  its  outline  will  furnish  convenient 
lines  for  reference. 

The  actual  work  of  executing  an  isometric  drawing  is  much 
less  than  one  would  suppose  from  a  perusal  of  these  notes,  which 
deal  with  theoretical  fundamentals;  with  a  thorough  knowledge 
of  these,  however,  and  with  some  little  practice,  many  short  cuts 
are  obvious  which  materially  shorten  the  process. 

Cavalier  Projection. 

12.  Introductory. — There   is   a   kind   of  drawing  closely  re- 
sembling isometric  drawing  and  isometric  projection,  which  is 


ISOMETRIC  DRAWING. 


29 


FIG.  30. 


FIG.  3I. 


30  ADVANCED  MECHANICAL  DRAWING. 

a  true  oblique  projection,  and  is  so  drawn  in  practice.  This 
class  of  drawing  is  called  "Cavalier  Projection";  it  is  more 
flexible  than  isometric  drawing,  it  portrays  three  faces  in  a  single 
view,  is  easily  and  readily  constructed,  and,  altogether,  is  well 
adapted  to  the  representation  of  small  machine  parts,  rectangular 
objects,  etc. 

13.  Theory. — In    Fig.    32    let     E-F-G-H    be    a    transparent 
plane  of    projection — say    a    portion  of    the  vertical  plane,  let 
X-Y  be  a  line  perpendicular  to  the  plane,  and  let  the  line  A-a-Y 
represent  a  line  of  sight  directed  at  the  plane  from  the  direction 
A,  and  making  an  angle  of  45°  with  the  plane;   now,  it  is  evident 
that  the  point  a  in  which  the  line  A  -a-  Y  pierces  the  plane  E-F-G-H 
is  the  projection  of  the  point  Y  on  the  plane;   it  is  also  evident 
that  the  point  X  of  the  perpendicular  X-Y  being  in  the  plane  is 
its  own    projection  on  the  plane;    therefore  the  line  a-X  is  the 
projection  of  the  perpendicular  X-Y,  and,  furthermore,  is  equal 
to  X-Y,  since  the  line  A-a-Y  makes  an  angle  of  45°  with  the 
plane. 

Assuming  the  point  of  sight  A  to  be  at  infinity,  as  in  ordinary 
mechanical  drawing,  the  lines  of  sight  all  become  parallel,  and 
the  projections  of  all  lines  perpendicular  to  the  plane  parallel 
to  a-X  and  equal  to  the  lines  themselves. 

Assuming  any  direction  for  the  point  of  sight,  as  B,  C,  D, 
etc.,  and  maintaining  the  45°  angle  with  the  plane,  it  is  seen 
that  the  line  a-X — the  projection  of  X-  Y — may  have  any  desired 
inclination,  the  true  length  of  the  projection  being  dependent 
upon  the  oblique  projection  of  45°,  and  the  angularity  upon  the 
assumed  direction  of  sight. 

14.  Application  of  the  Theory.— In    Fig.    33    let    E-F-G-H 
be  a  plane  of  projection,  let  the  dashed  figure   1-2-3-4-5-6-7-8 
represent  a  rectangular  block  with  its  face  1-2-3-4  in  the  plane 
E-F-G-H  and  its  side  faces,  3-4-8-7,  etc.,  perpendicular  to  the 
plane  of  projection,  and  let  the  lines  a,  a,  a,  etc.,  represent  lines 
of  sight  from  the  direction  A   directed  against  the  object  and 
making  an  angle  of  45°  with  the  plane  of  projection.     Now,  as 
in  the  previous  example,  it  is  evident  that  the  points  in  which 


ISOMETRIC  DRAWING. 


31 


B 


/ 

\\ 

/ 

/ 

+*>-T\ 

Line  of  Sight 

b 

"--%;.? 

Direction  B 

N 

^* 

X, 


FIG.  32. 


ADVANCED  MECHANICAL   DRAWING. 


the  lines  of  sight  to  the  corners  of  the  rear  base  pierce  the  plane 
of  projection  are  the  projections  of  the  corners  on  the  plane, 
and  when  properly  joined  by  right  lines  represent  the  projection 


FIG.  33. 

of  the  rear  base;  the  front  base,  or  face,  being  in  the  plane  is 
its  own  projection,  and  when  properly  connected  with  the  rear 
base  the  resulting  figure  (the  full-line  figure  1-2-3-4-5-6-7-8) 
represents  the  projection  of  the  block,  and  the  lines  of  the  pro- 
jection are  equal  to  the  lines  of  the  object. 


ISOMETRIC  DRAWING. 


33 


15.  Method  of  Procedure. — The  above  reduced  to  the  con- 
ventional method  of  representation  is  shown  by  Fig.  34,  and  is 
drawn  as  follows:  First  draw  the  face  which  is  its  own  projection 
to  any  desired  scale — full  size,  half-size,  etc. — exactly  as  in  ordi- 
nary orthographic  projection,  then,  having  decided  upon  a 
direction  of  sight,  draw  those  lines  which  represent  the  pro- 
jections of  lines  perpendicular  to  the  plane  of  projection  at  an 
inclination  (A°)  corresponding  to  the  direction  of  sight,  and 
equal  to  the  true  length  of  the  line  projected. 


FIG.  34. 


FIG.  35. 


16.  Flexibility  of   the   Art. — Fig.    35    shows    the   projection 
of  a  cube  viewed  from  four  directions,  and  illustrates  the  flexi- 
bility of  the  art.     The  angle  of  inclination  may  be  any  angle, 
but  usually  is  one  which  can  be  conveniently  drawn,  as  with 
the  45°  triangle,  the  60°  triangle,  etc.,  and  is  determined  in  prac- 
tice by  the  faces  of  the  object  to  be  pictured. 

17.  Practical    Examples. — This    method    of    representation 
being  so  like  an  isometric  representation  is  readily  understood 
and  acquired  by  one  well  versed  in  the  principles  of  isometry. 
As  in  isometric  drawing,  an  inclosing  box  may  be  used  and  the 
location  of  points  and  lines  obtained  by  means  of  offsets — plotting. 


34 


ADVANCED  MECHANICAL   DRAWING. 


The  method,  however,  has  two  advantages  over  the  isometric 
method  of  representation,  in  that  the  front  face  of  an  object 
is  drawn  a  true  orthographic  projection,  and  in  the  delineation 
of  circles  the  planes  of  which  are  parallel  with  the  plane  of  the 
front  face  of  the  object;  circles  other  than  these  are  drawn  as 
ellipses. 

Fig.  36  is  a  cavalier  projection  of  one-half  of  a  "split  brass" 
(a  bearing)  obtained  from  a  60°  direction  of  sight.     Note  the 


dimensioning  of  the  figure,  which  is  similar  to  that  used  in  iso- 
metric drawing,  then  the  construction  of  the  drawing,  which 
is  as  follows: 

In  cases  such  as  this,  where  the  object  is  symmetrical  with 
certain  center  lines,  it  is  well  to  draw  the  center  lines  first  and  use 
them  as  the  basis  of  construction,  as  in  ordinary  mechanical  draw- 
ing; therefore,  draw  the  center  lines  A' -A'  and  A-X  for  the  front 
face,  and  the  60°  center  line  A-F  for  the  top  face;  these  lines 


ISOMETRIC  DRAWING. 


35 


drawn,  draw  the  two  circles  of  the  front  face,  A- A  and  A'-A't 
then  locate  the  centers  B,  C,  D,  E,  and  F  on  the  line  of  centers 
(center  line)  A-F  in  accordance  with  the  dimensions  of  the 
object,  and  draw  the  other  circles  of  the  figure.  The  method 
of  drawing  the  other  lines  of  the  projection  is  similar  to 
that  used  in  isometric  drawing,  and  is  clearly  shown  in  the 
figure. 


FIG.  37. 


It  will  be  remarked  that  J"  fillets  are  drawn  exactly  as  in 
isometric;  this  feature  is  a  convenience  of  the  60°  direction  of 
sight,  and  does  not  occur  when  the  direction  of  sight  is  other 
than  this;  where  the  direction  is  15°,  30°,  45°,  etc.,  circles  and 
arcs  of  circles  appearing  in  either  the  top  or  side  planes  may  be 
drawn  by  drawing  the  major  and  minor  axes  of  the  ellipse,  the 
directions  for  which  are  known,  as  is  also  their  extent  (equal  to 


36  ADVANCED  MECHANICAL  DRAWING. 

the  diameter  of  the  circle,  or  double  the  radius  of  the  arc),  then 
the  ellipse  by  any  standard  method;  however,  with  the  axes  of 
the  ellipse  known,  it  is  common  practice  to  draw  a  rhombus 
and,  with  this  for  reference,  to  pencil  in  the  ellipse  or  arc  free- 
hand, and,  when  satisfactory,  to  ink  it  in  with  the  drawing  instru- 
ments. 

Screw-threads.  Since  the  real  advantage  of  cavalier  pro- 
jection over  isometric  drawing  is  in  the  delineation  of  circles,  it 
follows  that  the  representation  of  screw-threads  is  greatly  sim- 


B 


FIG.  38. 


plified.  Fig.  37  is  a  cavalier  projection  of  a  small  face-plate 
for  a  wood-turning  lathe,  and  illustrates  the  representation  of 
screw-threads,  the  arcs  representing  the  points  of  the  threads 
being  arcs  of  circles  (the  centers  for  which  are  all  on  the  center 
line  A-B)  and  all  parallel. 

The  representation  of  the  V's  of  the  thread  is  a  laborious 
and  time-consuming  construction,  and  can  be  expedited  by 
simply  indicating  the  thread  as  in  ordinary  mechanical  draw- 
ing, as  shown  by  Fig.  38,  the  distance  between  the  arcs  corre- 
sponding, approximately,  to  the  pitch  of  the  thread. 


ISOMETRIC  DRAWMG. 


37 


18.  Distortion. — A    reference   to  some  of  the  figures  shows 
an  unpleasant  distortion,  as  is  present,  also,  in  isometric  draw- 


FIG.  39. 

ings  of  certain  character;  where  the  drawing  will  permit  of  it, 
this  feature  may  be  eliminated  to  a  degree,  by  shortening  the 
width  or  length  of  the  projection,  as  shown  by  Fig.  39. 


CHAPTER  II. 

SHADOWS. 

19.  Introductory. — Without   light  and    shade    a   drawing   is 
merely  a  flat  outline.    A  simple  outline  drawing,  shade-  or  back- 
lined,  answers  for  usual  shop  purposes;   for  catalogue  and  show 
purposes  it  is  sometimes   desirable  to  have  a  drawing  depicting 
the  light  and  shade.     The  preparation  of  such  drawings  is  a 
trade  in  itself;    however,  the  engineer,  at  times,  may  desire  to 
produce  a  handsome,  shaded  drawing,  and  having  a  knowledge 
of  shading  to  convey  form   (cylindrical,  inclined,  concave   and 
convex  surfaces,  etc.),  he  may  enhance  his  work  by  the  addition 
of  shadows. 

It  is  the  purpose  of  these  notes  to  impart  a  working  knowledge 
of  the  finding  of  cast  shadows — a  practice  seldom  resorted  to 
in  ordinary  commercial  mechanical  drawing,  though  used  to 
some  extent  in  architectural  work. 

20.  Theory  of  Shadows. — The  rays  of  light  are  commonly 
assumed  as  emanating  from  the  sun,  and  coming  from  such  a 
distance  they  are  assumed  to  be  parallel  and  usually  at  45°  to 
the   planes  of  projection,  as  shown  by  Fig.  40.      These    rays, 
unobstructed,    illuminate   the    planes    of    projection;     however, 
should  a  ray  be  intercepted,  it  would  not  reach  the  planes  of  pro- 
jection  and   there   would   be   a   spot   thereon   unilluminated — a 
shadow.     That  is,  if  P,  Fig.  41,  be  a  point  in  space,  it  will  intercept 
a  ray  of  light  and  will  cast  a  shadow  on  the  plane  of  projection 
first  reached  by  the  intercepted  ray  if  unobstructed — the  shadow 
being,  clearly,  the  point  in  which  the  ray  would  pierce  the  plane. 

From  the  above  it  is  evident  that  to  find  the  shadow  of  a 

38 


SHADOWS. 


39 


point  one  has  but  to  pass  a  ray  of  light  through  it  and  find  the 
point  in  which  the  ray  pierces  the  planes  of  projection.  For  the 
purpose  of  finding  cast  shadows  the  planes  of  projection  are 
assumed  to  be  opaque  and  to  stop  the  rays  of  light;  from  this  it 


FIG.  40. 

is  seen  that  the  shadow  is  cast  on  that  plane  first  pierced  by  the 
light. 

To  find  the  shadow  cast  by  lines  and  objects  one  has  but  to 
find  the  shadow  of  a  number  of  points  and  then  join  the  shadow- 
points  in  the  proper  manner. 

Since  the  light  is  commonly  assumed  to  be  at  45°  with  the 
planes  of  projection,  it  is  evident  that  the  first  quadrant  is  the 
only  one  in  which  the  rays  will  cast  shadows  on  both  planes; 


ADVANCED  MECHANICAL   DRAWING. 


and  since  these  are  both  visible  here,  it  is  usual  to  assume  the 
object  situated  in  the  first  quadrant. 

21.  The  Shadow  of  a  Point. — As  already  stated,  to  find  the 
shadow  of  a  point,  pass  a  ray  of  light  through  the  point  and  find 
the  point  in  which  it  pierces  the  planes  of  projection — the  shadow 
being  the  point  of  piercing  the  plane  first  reached. 

It  will  be  remarked  that  this  is  a  practical  application  of 
the  elementary  problem  in  the  Descriptive  Geometry,  "Find 


>tol- 


-^•5fC\! 

ST^TP'       -/       \N 


^ 


Horizontal  Plane. 


FIG.  41. 

the  point  in  which  a  line  pierces  the  planes  of  projection."  The 
procedure  is  a  simple  one,  but,  as  it  is  the  basis  of  all  shadow- 
work,  the  student  should  understand  it  thoroughly;  he  must 
remember  that  a  point  has  two  projections;  also,  that  a  line  has 
two  projections,  and  that  in  passing  a  ray  of  light  through  a  point 
by  the  convention  used  in  drawing,  he  must  draw  the  vertical 
projection  of  the  ray  through  the  vertical  projection  0}  the  point, 
and  the  horizontal  projection  of  the  ray  through  the  horizontal 
projection  of  the  point,  and  that  the  point  on  the  plane  of  pro- 
jection first  reached  by  the  ray  is  the  shadow-point. 


SHADOWS. 


To  illustrate  the  above,  consider  Fig.  42:  A  shows  the  pro- 
jections p  and  pf  of  a  point  P  in  space  through  which  a  ray  of 
light  is  passed;  this  ray  is  seen  to  pierce  the  vertical  plane 
first,  hence  the  shadow  of  P  falls  on  the  vertical  plane.  B  shows 


FIG.  42. 

tne  projections  of  a  point  P  so  situated  with  reference  to  the 
planes  of  projection  that  the  shadow  falls  on  the  horizontal  plane. 
22.  The  Shadow  of  a  Right  Line. — Since  a  line  is  made  up 
of  points,  any  two  points  in  the  line  may  be  taken  (provided  the 
same  points  are  shown  in  the  two  projections,  that  is,  provided 
the  points  are  properly  projected)  and  the  shadow  of  these  points 
found  as  directed  above,  and  the  shadow  of  the  line  obtained 


42  ADVANCED  MECHANICAL  DRAWING. 

by  joining  the  shadow-points  by  a  right  line.  If  the  line  be  a 
definite  line,  the  two  points  taken  are,  clearly,  the  two  extremes 
of  the  line. 

The  shadow  falling  on  one  plane  only.  A,  Fig.  43,  shows 
the  projections  of  a  definite  line  M-N,  through  the  extremes 
of  which  rays  of  light  are  passed  and  the  shadow  (M'-N') 
found  in  accordance  with  the  above  explanation.  It  will  be 


FIG.  43. 

noted  that  both  of  the  rays  used  pierce  the  vertical  plane  first, 
and  that  of  a  consequence  the  shadow  falls  on  this  plane. 

The  shadow  falling  on  both  V  and  H.*  When  a  line  is  so 
situated  that  its  shadow  falls  on  both  of  the  planes  of  projection, 
it  is  necessary  to  find  the  points  in  which  the  rays'  pierce  both 
V  and  H,  the  points  on  the  planes  first  reached  »by  the  rays  not 
being  sufficient,  as  is  apparent  from  an  inspection  of  B,  Fig.  43. 
Here  is  depicted  the  projections  of  a  line  M-N,  through 
the  extremes  of  which  the  projections  of  rays  of  light  are 
drawn;  it  is  seen  that  the  ray  through  the  point  M  pierces  the 
horizontal  plane  first,  and  that  the  point  M  on  this  plane  is  the 
shadow  of  the  If  extreme  of  the  line.  Now,  it  is  evident  that  to 

*  In  this  discussion  the  letter  H  stands  for  the  horizontal  plane  of  projection 
and  the  letter  V  for  the  vertical  plane  of  projection. 


SHADOWS. 


43 


find  the  shadow  of  the  line  on  the  horizontal  plane,  a  second 
shadow-point  must  be  found  on  it,  and,  naturally,  one  finds  the 
shadow  of  the  other  extreme  of  the  line.  The  ray  through  this 
point  is  seen  to  pierce  the  vertical  plane  first,  then  to  continue 
on  and  pierce  the  horizontal  plane  in  the  second  quadrant.  Join- 
ing the  two  shadow-points  on  the  horizontal  plane  gives  the  shadow 
of  the  line  on  H,  but  since  the  first  quadrant  only  is  considered, 
that  part  of  the  connecting  line  which  is  in  the  first  quadrant 
is  the  only  part  of  the  shadow  visible. 

Having  the  shadow  of  the  N  extreme  on  the  vertical  plane, 

\m' 


A  B 

FIG.  44. 

the  shadow  of  the  line  may  be  completed  by  joining  the  point 
in  which  the  shadow  on  the  horizontal  plane  crosses  the  ground- 
line  with  this  point,  or,  by  finding  the  shadow  of  the  M  extreme 
on  the  vertical  plane  (this  is  seen  to  be  in  the  fourth  quadrant) 
and  joining  this  point  with  the  V  shadow  of  the  N  extreme.  If 
this  latter  method  is  used  the  H  and  V  shadows  of  the  line  should 
cross  at  the  ground- line. 

The  shadow  of  a  line  when  parallel  to  one  of  the  planes  of 
projection.  A,  Fig.  44,  shows  the  projections  of  a  line  M-N 
which  is  parallel  to  the  horizontal  plane  of  projection,  together 
with  its  shadow,  M-N,  on  this  plane  (the  line  is  so  assumed  that 
its  shadow  falls  entirely  on  H).  Now,  since  the  line  is  parallel 


44 


ADVANCED  MECHANICAL  DRAWING. 


to  H,  and  since  the  rays  of  light  are  assumed  to  be  parallel,  it 
follows  that  the  length  of  the  lines  m-M  and  n-N  are  equal  and 
the  line  M-N  therefore  parallel  and  equal  to  m-n. 

B,  Fig.  44,  shows  the  projections  of  a  line  which  is  perpen- 
dicular to  H,  and  hence  parallel  to  F,  together  with  its  shadow, 
which  is  seen  to  fall  on  both  H  and  V  (the  line  being  so  assumed). 
The  method  of  finding  the  shadow  is  the  same  method  as  is 
described  on  page  42. 

An  inspection  of  the  shadow  shows  that  that  part  of  it  falling 
on  H  is  in  the  direction  of  the  H  projection  of  the  rays  of  light, 
and  that  portion  falling  on  V  is  parallel  to  the  V  projection  of 
the  line. 


FIG.  45- 

The  above  examples  demonstrate  two  points  which  should 
be  well  fixed  in  mind,  as  a  knowledge  of  them  will  greatly  expedite 
the  work  of  finding  cast  shadows.  These  points  are: 

(1)  The  shadow  of  a  line  on  a  parallel  plane  is  parallel  and 
equal  to  the  line. 

(2)  The  shadow  o)  a  line  on  a  plane  to  which  it  is  perpen- 
dicular lies  in  the  direction  of  the  projection  of  the  rays  on  that 
plane. 

23.  The  Shadow  of  a  Curved  Line.— Fig.  45  shows  the 
projections  of  a  curved  line  M-N,  together  with  its  shadow 


SHADOWS.  45 

M'-i'-2-3~N.  The  line  is  divided  into  a  number  of  points, 
the  shadow  of  each  point  found,  and  the  shadow-points  joined 
with  a  curved  line.  Therefore, 

To  find  the  shadow  of  a  curved  line,  find  the  shadow  of  a  number 
of  points  in  the  line  and  join  the  shadow-points  with  a  curved  line. 

Since  the  shadow  of  a  line  on  a  parallel  plane  is  parallel 
and  equal  to  the  line,  it  follows  that 

The  shadow  of  a  plane  curve  on  a  parallel  plane  is  parallel 
and  equal  to  the  curve. 

For  example: 

To  find  the  shadow  of  a  circle  on  a  plane  to  which  its  plane  is 
parallel,  find  the  shadow  of  the  center  oj  the  circle,  and  with  this 
point  as  a  center  and  a  radius  equal  to  the  radius  oj  the  circle  draw 
a  circle;  this  circle  will  be  the  shadow  oj  tlie  given  circle. 

24.  The  Shadow  of  Solids.  Plane  Surfaces. — A,  Fig.  46, 
shows  the  projections  of  a  small,  rectangular  block,  the  eight 
corners  of  which  are  numbered  and  the  shadow  cast  by  each 
corner  shown.  The  shadow  cast  by  the  lines  or  edges  of  the 
object  are  shown  by  the  right  lines  joining  the  shadows  cast 
by  the  corner  points;  thus  the  shadow-line  8-4  is  the  shadow  of 
the  edge  8-4,  etc. 

An  inspection  of  the  figure  shows  conditions  such  that  the 
shadow  falls  entirely  on  the  horizontal  plane;  also,  that  the  top 
and  bottom  planes  or  bases  of  the  block  are  parallel  to  H,  and 
that  the  lines  or  edges  joining  the  two  bases  are  perpendicular 
to  H.  It  is  interesting  to  note,  then,  that  the  shadows  of  the 
lines  of  the  bases  are  parallel  and  equal  to  the  same  lines  on  the 
object,  and  that  the  shadow-lines  joining  the  shadow  bases  lie 
in  the  direction  of  the  H  projection  of  the  rays  of  light. 

A  further  inspection  of  the  figure  shows  a  number  of  the 
shadow-points  and  lines  to  fall  within  the  outline — the  limits — 
of  the  shadow.  This  presents  the  real  problem  in  shadows, 
i.e.,  to  find  the  outline  of  the  shadow  cast  by  an  object  without 
finding  the  shadow  of  all  of  its  points  and  lines.  To  find  the 
shadow  of  every  point  and  line  of  an  object,  many  of  which  fall 
within  the  limits  of  the  shadow,  is,  obviously,  both  time-con- 


46  ADVANCED  MECHANICAL   DRAWING. 

suming  and  laborious  and  to  be  avoided  in  so  far  as  practicable. 
To  this  end,  then,  the  student  must  be  able  to  read  his  drawing — 
the  presentation  of  the  object  by  the  two  projections — in  such 
a  manner  as  to  conceive  of  the  object  as  occupying  the  position 
of  the  projection  on  the  paper,  to  actually  stand  out  from  the 

The  Object 


FIG.  46. 

surface  as  if  the  object  rested  there,  and  with  the  object  thus  in 
mind  carefully  study  it  and  select  those  points  and  lines  which 
will  cast  the  desired  outline.  These  points  and  lines  known, 
it  is  then  a  very  simple  matter  to  find  the  shadow,  as  witness 
B,  Fig.  46. 

Having  considered  A  of  the  same  figure,  it  is  known  that  the 
lines  5-8  and  5-6  of  the  lower  base,  the  edges  6-2  and  8-4  of  the 
side  faces,  and  the  lines  2-3  and  3-4  of  the  upper  base  cast  the 
outline  of  the  shadow  of  the  block.  B  shows  a  new  position 
of  the  object  with  reference  to  the  planes  of  projection,  but  not 


SHADOWS. 


47 


with  reference  to  the  rays  of  light,  and  hence  the  same  lines 
6-2,  8-4,  etc.,  will  cast  the  required  outline.  This  known,  note 
the  method  of  finding  the  shadow  (B,  Fig.  46).  The  correctness 
of  the  shadow  may  be  recognized  and  the  labor  of  execution 
further  reduced  if  one  will  remember  that  the  shadow  of  a  line 
on  a  parallel  plane  is  parallel  and  equal  to  the  line;  thus,  having 
found  the  shadow-point  8,  the  shadow-line  8-5  may  be  drawn 
parallel  and  equal  to  the  edge  8-5  of  the  block,  the  shadow-line  5-6 
may  be  drawn  parallel  and  equal  to  the  edge  5-6  of  the  block,  etc. 
The  shadow  on  the  object.  Fig.  47  illustrates  an  object  of 


12 


9       10 

J2 

/4 

3         4 

r       e 

FIG.  47. 

such  form  and  position  with  reference  to  the  planes  of  projection 
that,  in  addition  to  the  shadow  cast  on  the  planes  of  projection, 
it  casts  a  shadow  on  itself. 

The  shadow  cast  on  the  planes  is  found  as  in  the  preceding 
example,  and  as  shown  by  the  drawing  (note  the  shadow  cast  by 


48  ADVANCED  MECHANICAL  DRAWING. 

those  lines  which  are  parallel  and  perpendicular  to  F  and  H). 
The  shadow  cast  by  the  object  on  itself  is  found  as  follows : 

As  already  explained  (page  45)  the  drawing  must  be  studied 
and  those  lines  selected  which  will  cast  the  outline  of  the  shadow. 
Such  an  inspection  of  the  figure,  then,  shows  that  the  lines  5-4 
and  4-10  are  the  only  lines  concerned  in  the  shadow,  and  that 
the  shadow  will  fall  on  the  plane  5-11-12-6;  note,  also,  that  this 
plane  is  parallel  to  H  and  that  the  line  5-4  is  perpendicular  and  the 
line  4-10  parallel  to  this  plane.  Now  the  line  5-4  being  perpen- 
dicular to  the  plane  5-11-12-6,  its  shadow  will  lie  in  the  direction 
of  the  projection  of  rays  on  this  plane,  and  the  point  5  being  in 
the  plane  will  be  its  own  shadow;  to  find  the  shadow  of  point 
4,  pass  a  ray  of  light  through  it  and  find  the  point  .in  which  it 
pierces  the  plane  5-11-12-6;  this  point  is  seen  to  be  point  4  on 
the  plane  5-11-12-6,  and  the  shadow  of  the  line  5-4  to  be  the  line 
joining  the  point  5  and  this  point.  (Note  how  the  shadow  of 
point  4  is  found:  how  the  V  projection  of  a  ray  through  the  V 
projection  of  the  point  strikes  the  V  projection  of  the  plane  at 
X,  and  the  projection  of  this  point  to  the  horizontal  projection 
of  the  ray  through  the  horizontal  projection  of  point  4;  that  is, 
how  the  V  projection  of  the  plane  5-11-12-6  serves  as  a  ground- 
line,  as  it  were.)  The  shadow-point  4  determined,  and  knowing 
that  since  the  line  4-10  is  parallel  to  the  plane  5-11-12-6,  its  shadow 
thereon  will  be  parallel  and  equal  to  the  line  itself,  the  shadow- 
of  this  line  may  be  found  by  drawing  a  line  through  the  shadow- 
point  4  parallel  to  the  edge  4-10,  and  of  a  length  equal  to  the 
length  of  the  edge;  such  a  length,  however,  carries  the  shadow 
ofl  of  the  object  and  onto  the  horizontal  plane,  and  falling  there 
within  the  limits  of  the  shadow  is  disregarded. 

Fig.  48  shows  a  second  example  of  a  shadow  falling  on  the 
object.  The  method  of  finding  the  shadow  on  the  planes  of 
projection  and  on  the  horizontal  projection  of  the  object  is  the 
same  as  has  already  been  described  and  illustrated,  and  as  is 
shown  by  the  figure.  However,  the  object  is  of  such  form  and 
position  with  reference  to  the  light  that  a  new  feature  is  introduced 
in  that  a  shadow  is  cast  by  the  object  on  itself  on  a  plane  which 


SHADOWS. 


49 


is  visible  in  its  vertical  projection.  This  shadow  is  seen  to  be 
the  shadow  of  the  lines — edges — 2-6  and  6-5  on  the  plane  1-2-3-4. 
These  lines  being  perpendicular  and  parallel  respectively  to  the 
plane,  the  shadow  is  found  as  in  the  preceding  example,  and 
as  is  shown  by  the  drawing. 


The  Object 


FIG.  48. 


Single-curved  surfaces.  Fig.  49  illustrates  the  plan  and 
elevation  of  a  section  of  a  hollow  cylinder  and  introduces  the 
finding  of  cast  shadows  on  single-curved  surfaces. 

A  study  of  the  figure  shows  that  it  casts  a  shadow  on  itself 
and  on  the  planes  of  projection ;  also,  that  the  limits  of  the  shadow 
on  the  planes  is  cast  by  elements  C-C,  A- A,  and  B-B,  and 
the  curve  B-A  of  the  top  base  of  the  cylinder,  and  the  shadow  on 
itself  by  the  element  C-C,  and  the  lines  C-E  and  E-D  of  the  upper 
base.  The  shadows  of  the  straight  lines  are  found  as  already 
described,  and  those  of  the  curved  lines  as  directed  on  page 
44,  and  as  shown  on  the  drawing  by  the  points  a  and  b. 

Fig.  50  is  a  second  example  of  a  single-curved  surface  and 
illustrates  a  method  for  finding  the  shadow  of  a  right  line  on  the 
surface. 


5° 


ADVANCED  MECHANICAL   DRAWING. 


The  shadow  on  the  planes  of  projection  is  found  in  the  usual 
way,  and  the  shadow  on  the  figure  as  follows: 

A  study  of  the  object  shows  that  the  shadow  on  the  object 
is  cast  by  the  projection  of  the  top  or  cap,  and  that  the  shadow 
will  show  in  the  vertical  projection  of  the  figure.  Now  it  is 
evident  that  when  viewing  the  vertical  projection  one  sees  but 


-  A 


FIG.  49. 

one-half  of  the  figure,  this  half  being  that  half  in  front  of  a  vertical 
plane  parallel  with  V  passing  through  the  center  of  the  object. 
The  point  a,  then,  in  the  line  1-8  is  the  first  point  on  the  left  to 
cast  a  visible  shadow.  To  find  the  shadow  cast  by  the.  line  0-8, 
divide  it  into  a  number  of  points,  as  b,  find  the  shadows  of  these 
points,  and  join  them  with  a  curved  line,  as  shown;  the  shadow 
of  the  line-edge  8-7  is  obtained  in  a  similar  manner. 

A  further  inspection  of  the  figure  shows  that  the  ray  of  light 


SHADOWS. 


through  the  point  d  of  the  line  or  edge  7-6  is  tangent  to  the  single- 
curved  surface — the  base  of  the  figure — at  the  point  k,  and  is  the 
last  point  on  the  right  to  cast  a  shadow  on  the  object,  the  remainder 
of  the  line  7-6,  d-6,  casting  its  shadow  on  the  horizontal  plane. 


FIG.  50. 

(It  is  interesting  to  note  that  this  shadow  is  parallel  and  equal  to 
the  line  d-6.)  Now,  the  ray  d-k  being  the  tangent  ray  it  is  obvious 
that  all  of  that  portion  of  the  surface  beyond  the  element  through 
the  point  of  tangency,  k,  is  in  the  shadow,  and  is  so  shown  in 
the  vertical  projection. 

Double-curved  surfaces.      Fig.  51  represents  the  projections 
of  a  block  so  hollowed  out  as  to  present  a  surface  part  of  which 


52 


ADVANCED   MECHANICAL   DRAWING. 


is  of  single  curvature  and  part  of  double  curvature,  and  is  typical 
of  the  architectural  niche  designed  to  house  a  statue. 

The  drawing  shows  the  shadow  within  the  niche  only;  it 
is  found  as  follows: 

The  shadow  on  the    single-curved    portion  of  the  recess  is 


FIG.  51. 

found  as  has  already  been  described,  and  as  is  shown  by  the 
drawing.  To  find  the  shadow  on  the  double- curved  surface, 
pass  a  series  of  vertical  projecting  planes  through  the  surface 
and  parallel  with  H,  as  shown  by  the  horizontal  lines  o'-i4', 
i'-i3',  2/-i2/,  etc.,  of  the  vertical  projection.  These  planes  will 
intersect  the  surface  in  semicircles  which  are  visible  on  the  H 
projection  as  the  curves  0-14,  1-13,  2-12,  etc.;  next,  through 
any  point  casting  a  shadow  on  the  surface,  as  point  6,  pass 
a  ray  of  light,  then  through  the  ray  pass  its  horizontal  projecting 


SHADOWS.  53 

plane,  and  by  projecting  from  the  points  of  intersection  of  the 
H  trace  of  this  plane  with  the  H  traces  of  the  above  auxiliary 
planes  to  the  V  traces  of  the  auxiliary  planes  find  the  V  trace 
of  the  plane  of  rays  on  the  surface ;  this  trace  is  the  curve  6'-F'. 
Now  it  is  evident  that  the  shadow  of  the  point  6  will  lie  in  the 
trace  of  the  plane  of  the  ray — the  curve  6'-F',  also,  that  the  shadow 
will  lie  in  the  projection  of  the  ray,  therefore,  the  intersection  of 
the  line  6'-F' — the  V  projection  of  the  ray — with  the  curve 
6'-F' — the  trace  of  the  plane  of  the  ray  on  the  surface — is  the 
required  shadow.  The  shadow  is  completed  by  selecting  a 
number  of  points  and  finding  their  shadows  in  a  similar  manner, 
then  joining  these  shadows  by  a  curved  line,  as  shown. 

The  feature  here  introduced  is  the  use  of  a  plane  of  rays, 
and  is  to  be  employed  whenever  the  surface  is  such  that  the  two 
projections  of  a  ray  are  not  sufficient  to  indicate  the  point  in 
which  the  ray  pierces  the  surface. 

Fig.  52  illustrates  the  shadow  cast  by  a  sphere.  The  shadow 
may  be  found  by  locating  the  great  circle  of  contact  of  the  rays 
of  light  by  the  method  used  in  the  dome  of  the  niche  in  the  previous 
example,  or  as  follows: 

Pass  a  cylinder  of  rays  about  the  sphere:  this  will  define  the 
great  circle  of  contact,  and  this  found,  select  a  number  of  points 
in  it,  find  their  shadows,  then  join  these  with  a  curved  line. 

In  executing  the  above  analysis,  one  meets  with  a  practical 
application  of  the  problem  in  Descriptive  Geometry,  "Pass  a 
circle  through  three  points,"  as  witness  the  figure: 

It  is  obvious  that  the  horizontal  projection  corresponds  with 
a  horizontal  section  of  the  sphere,  vertically  represented  as  the 
line  X-Y,  and  that  the  vertical  projection  corresponds  with  a 
vertical  section  of  the  sphere  horizontally  projected  as  the  line 
M-N]  it  is  evident,  then,  that  the  tangent-points  A  and  B  defined 
by  the  horizontal  projection  of  one  set  of  limiting  rays  of  light, 
and  the  tangent-points  E  and  D  defined  by  the  vertical  projection 
of  a  second  set  of  limiting  rays,  may  be  readily  projected,  and 
will  represent  the  projections  of  points  in  the  great  circle  of  contact 
of  the  cylinder  of  light  with  the  sphere. 


54 


ADVANCED  MECHANICAL  DRAWING. 


With  four  points  of  the  circle  of  contact  known,  it  is  an  easy 
matter  to  pass  a  plane  through  any  three  of  them,  to  revolve  the 
plane  of  the  points  into  coincidence  with  one  of  the  planes  of 
projection,  then,  while  in  this  position,  to  pass  a  circle  through 


N 


FIG.  52. 

the  points,  and  then  to  revolve  the  circle  back  to  the  original 
position  of  the  points;  the  projections  of  the  circle  will  then 
represent  the  projections  of  the  great  circle  of  contact  of  the 
cylinder  of  light.  The  circle  of  contact  defined,  the  shadow 
on  the  object  is  obvious,  and  the  shadow  on  the  planes  found  by 
finding  the  shadows  cast  by  a  number  of  points  in  the  circle  of 
contact,  then  joining  these  shadow-points  with  a  curved  line. 


SHADOWS.  55 

25.  Remarks. — Since,  the  ordinary  engineer  draughtsman  is 
so  little  concerned  with  the  finding  of  cast  shadows,  it  is  thought 
that  this  brief  discussion  of  the  subject  is  sufficient ;  the  examples 
given  are  typical  ones,  and  carefully  studied  show  that  there 
is  but  one  principle  involved,  i.e.,  that  principle  of  orthographic 
projection  enabling  one  to  find  the  points  in  which  a  given  line 
pierces  a  plane  or  surface  of  projection.  The  subject,  however, 
is  of  importance  in  the  training  of  the  student  engineer  in  that 
it  shows  a  practical  application  of  the  principles  of  descriptive 
geometry,  and  affords  excellent  practice  in  the  reading  of  drawings. 


CHAPTER  III. 
PERSPECTIVE. 

26.  Definition. — Perspective  drawing,  or  Linear  Perspective, 
commonly  called  "Perspective,"   is  the   art  of  representing  an 
object  or  objects  on  paper  or  other  plane  surface  in  such  a  manner 
as  to  present  the  object  as  it  would  appear  when  viewed  from 
a  definite  viewpoint. 

27.  Perspective  and  Mechanical  Drawing  Compared. — Per- 
spective  differs   from   mechanical  drawing,  which   presents   an 
object  in  detail — each  face  or  side  separately,  and  as  it  really 
is,  and  not  as  it  appears  to,  the  eye — in  that  it  presents  the  object 
as  a  whole,  showing  several  faces  or  sides  in  a  single  drawing,  and 
as  it  would  appear  if  viewed  from  a  given  standpoint. 

28.  Mechanical    and  Free-hand    Perspective. — These  terms 
are    used    relatively.       By    " mechanical"  perspective   is   meant 
that  perspective  drawing  of  an  object  which  is  drawn — said  to 
be   " found" — from  the  mechanical  drawings  of  it;  that  is,  to 
execute  a  mechanical  perspective  of  an  object,  one  has  to  first 
prepare  mechanical  drawings  of  the  faces  or  sides  it  is  desired  to 
picture  and  then  use  these  drawings  to  find  the  perspective. 

By  free-hand  perspective  is  meant  that  perspective  drawing 
which  is  drawn  directly  from  the  object  or  scene;  that  is,  the 
artist,  or  draughtsman,  prepares  this  perspective  by  simply  look- 
ing at  the  thing  to  be  pictured  and  then  draws — free-hand — 
his  conception  of  it.  The  work  of  the  newspaper  artist,  the 
landscape-  and  portrait-painter,  etc.,  are  examples  of  "  free-hand 
perspective." 

56 


PERSPECTIVE.  57 

The  student  of  mechanical  drawing  is  primarily  concerned 
with  mechanical  perspective,  and  it  is  this  class  only  which  is 
here  discussed. 

29.  Perspective   as  Applied  by  the   Engineer. — The   art   of 
perspective   drawing   is   of   minor  importance   to   the   engineer, 
since  his  conceptions  are  best  expressed  by  simple  mechanical 
drawings.     An  exception,  however,  is  met  with  in  the  case  of 
the  architectural  engineer.     To  him  perspective  drawing  is  equally 
as  important  as  mechanical  drawing,  since  he  "pictures"  his 
proposed  work. 

While  it  is  true  that  the  art  is  mostly  applied  by  the  architec- 
tural engineer,  it  is  equally  true  that  all  engineers,  sometime, 
find  it  desirable  to  picture  a  conception,  and  therefore  it  is  well 
to  have  a  working  knowledge  of  the  principles  of  the  art. 

The  following  examples  and  remarks,  then,  are  not  designed 
to  produce  expert  perspective  draughtsmen,  but  as  the  funda- 
mentals are  given,  when  these  are  understood  one  may  become 
quite  expert  in  the  art  with  practice. 

30.  Theory  of  Perspective. — If  an  object  is  viewed  from  a 
finite  point  of  sight,  the  lines  of  sight  will  converge  at  the  eye 
(one  sees  with  two  eyes  but  in  perspective  a  single  point  of  sight 
is  assumed,  and  to  make  the  analogy  correct  the  observer  is 
supposed  to  close  one  eye),  and  if  they  be  intersected  by  a  plane 
—usually  assumed  to  be  between  the  object  and  the  point  of 
sight  for  reasons  explained  later — and  the  points  in  which  the 
several  lines  of  sight  pierce  it  be  properly  connected,  a  drawing 
is  obtained  which  represents  the  object,  decreased  in  size,  exactly 
as  it  appears  to  the  observer.     (Fig.  53.) 

The  intersecting  plane  assumed  in  practice  is  the  vertical 
plane  of  orthographic  projection,  being  assumed  because  of  its 
position,  which  enables  one  to  place  objects  to  be  pictured  with 
a  large  number  of  their  principal  lines  either  parallel  or  perpen- 
dicular to  the  plane,  thus  expediting  the  work  of  constructing 
the  perspective;  it  is  called  the  "plane  of  the  picture  or  picture 
plane,"  while  the  orthographic  projection  of  the  point  of  sight 
on  this  plane  is  called  the  "principal  point  of  the  picture." 


ADVANCED  MECHANICAL  DRAWING. 


PERSPECTIVE.  59 

From  the  above  it  is  seen  that  the  theory  of  the  art  is  very 
simple,  that  is,  to  find  the  perspective  of  a  point  one  has  but  to 
find  the  point  in  which  the  line  of  sight  to  the  point  pierces  the 
picture  plane,  and  since  the  picture  plane  is  assumed  to  be  the 
vertical  plane  of  orthographic  projection,  and  the  point  of  sight 
and  the  given  point  are  assumed  by  their  two  projections,  and 
the  projections  of  the  line  of  sight  as  lines  joining  the  projections 
of  the  two  points,  the  entire  procedure  becomes  a  practical  appli- 
cation of  the  elementary  problem  in  Descriptive  Geometry,  "Find 
the  point  in  which  a  line  pierces  the  planes  of  projection." 

31.  The  Perspective  of  a  Point. — With  the  theory  of  per- 
spective well  in  mind,  the  finding  of  the  perspective  of  a  point 
should  be  a  very  easy  matter.  However,  the  writer  has  observed 
that  students  have  some  difficulty  in  applying  the  theory  even 
here,  due  to  the  fact  that  they  forget  that  a  point  or  line  is  as- 
sumed by  its  two  projections.  In  the  example  to  follow,  note 
that  the  point  of  sight  has  a  horizontal  and  a  vertical  projection, 
that  the  given  point  has  two  like  projections,  and  that  the  horizontal 
projection  of  the  line  of  sight  is  a  line  joining  the  horizontal  pro- 
jection of  the  point  of  sight  and  the  horizontal  projection  of  the 
given  point;  also,  that  the  vertical  projection  of  the  line  of  sight 
is  a  line  joining  the  two  vertical  projections.  (Do  not  join  a 
horizontal  and  a  vertical  projection,  or  vice  versa.) 

Example.  In  Fig.  54  let  P  be  a  point  in  the  second  quadrant, 
and  let  p  be  its  horizontal  projection  and  {/  its  vertical  projection, 
and  let  5  be  the  point -of  sight  situated  in  the  first  quadrant,  and 
let  s  be  its  horizontal  projection  and  ^  be  its  vertical  projection. 

By  definition  a  line  of  sight  is  a  line  joining  a  point  and  the 
point  of  sight,  from  which  it  is  seen  that  the  line  S-P  represents 
the  line  of  sight,  and  the  lines  s-p  and  sf-f^  its  horizontal  and 
vertical  projection  respectively.  This  line  of  sight  pierces  the 
vertical  or  picture  plane  at  P',  which  point,  according  to  the  theory 
of  perspective,  is  the  perspective  of  point  P. 

To  check  the  above,  and  to  apply  the  principles  of  geometry, 
consider  the  two  projections  of  the  line  of  sight  and  note  that 
the  line  pierces  the  vertical  plane  at  P'. 


6o 


ADVANCED  MECHANICAL  DRAWING. 


To  discuss  the  example  on  practical  and  familiar  ground, 
consider  Fig.  55*:    The  point  P  being  assumed  in  the  second 


As 


n\ 


FIG.  54- 


FIG.  55. 

quadrant  has  its  two   projections    above   the   ground-line  G-G, 
and  the  point  of  sight  being  in  the  first  quadrant  has  its  vertical 

*  The  student  will  remark  a  violation  of  the  conventions  of  second  quadrant 
projection  in  this  and  other  figures  in  this  chapter,  in  that  the  vertical  projection 
of  lines  is  invisible  and  should  be  represented  by  a  dashed  line.  It  was  thought, 
however,  that  such  notation  would  result  in  confusion  in  certain  cases,  and  that 
used  is  arbitrarily  taken  for  the  occasion  in  the  belief  that  it  makes  the  figures 
more  clear. 


PERSPECTIVE.  6 1 

projection  above  the  ground-line  and  its  horizontal  projection 
below.  Joining  the  horizontal  projection  of  the  point  of  sight 
with  the  horizontal  projection  of  the  given  point  gives  the  hori- 
zontal projection  of  the  line  of  sight,  and  joining  the  vertical 
projection  of  the  point  of  sight  with  the  vertical  projection  of  the 
given  point  gives  the  vertical  projection  of  the  line  of  sight; 
the  line  thus  determined  is  found  to  pierce  the  vertical  plane 
at  P',  the  perspective  of  the  point. 

32.  The  Perspective  of  a  Right  Line. — To  find  the  persepctive 
of  a  right  line  one  has  but  to  find  the  perspective  of  any  two  points 
in  the  line  and  then  join  these  two  perspectives  with  a  right  line. 
If  the  given  line  be  a  definite  line  the  two  points  taken  are  the 
extremes  of  the  line. 

To  illustrate,  consider  Fig.  56,  which  shows  a  line  M-N  in 
the  second  quadrant,  together  with  its  projections  m-n  (horizontal) 
and  m'-nr  (vertical);  also  a  point  of  sight,  5,  situated  in  the  first 
quadrant  together  with  its  two  projections  s  (horizontal)  and  sf 
(vertical).  The  given  line  is  a  definite  one,  and  to  find  its  per- 
spective, find  the  perspective  of  the  extremes  M  and  TV  and  join 
them  by  a  right  line.  By  section  31  these  perspectives  are  found 
to  be  M'  and  AT',  and  the  perspective  of  the  line  to  be  M'-N'. 

Fig.  57  illustrates  the  conventional  orthographic  projection 
of  the  above  example. 

33.  The  Perspective  of  a  Curved  Line.  -  -  The  perspective 
of  a  curved  line  is  found  by  finding  the  perspective  of  a  number 
of   its  points  and  then  joining  these  perspectives  by  a  curved 
line. 

34.  Why  Objects  are  Assumed  in  the  Second  Quadrant.— The 
student  may  have  remarked  that  in  both  of  the  foregoing  ex- 
amples the'  thing  given  is  situated  in  the  second  quadrant  and 
the  point  of  sight  in  the  first  quadrant.     This  is  the  usual  practice, 
the  reason  for  which  is  as  follows:    In  Fig.  53  the  lines  of  sight 
are  seen  to  converge  at  the  point  of  sight,  and  it  is  clearly  evident 
that  as  the  position  of  the  picture  plane  is  changed,  the  size  of 
the  picture  or  perspective  is  changed,  becoming  smaller  as  the 
plane  is  moved  toward  the  point  of  sight  and  larger  as  it  is  moved 


62 


ADVANCED  MECHANICAL  DRAWING. 


c/>O— 


FIG.  57. 


64  ADVANCED  MECHANICAL  DRAWING. 

toward  the  object.  It  is  evident  also  that  so  long  as  the  inter- 
secting plane  is  between  the  object  and  the  point  of  sight  the 
perspective  will  be  smaller  than  the  object,  and  that  if  the  plane 
be  placed  beyond  the  object  the  perspective  will  be  larger  than 
the  object. 

In  nearly  every  case  a  picture  which  is  smaller  than  the  object 
is  desired,  hence  the  assumption  of  the  object  in  the  second 
quadrant  and  the  point  of  sight  in  the  first  quadrant,  or,  in  other 
words,  the  assumption  of  the  picture  plane  between  the  object 
and  the  point  of  sight. 

35.  The  Perspective  of  an  Indefinite  Right  Line. — To  find  the 
perspective  of  any  right  line,  proceed  as  in  section  32,  or,  since 
all  right  lines  may  be  considered  as  indefinite  right  lines  (definite 
right  lines  may  be  produced  or  extended)  a  second  method  for 
finding  the  perspective  of  the  line  may  be  used.  This  method 
is  as  follows: 

If  a  line  is  of  indefinite  length  it  will  pierce  the  picture  plane 
at  some  point,  unless,  of  course,  the  line  is  parallel  to  the  plane. 
This  point  is  clearly  a  point  in  the  perspective  of  the  line.  Hav- 
ing, then,  one  point  in  the  perspective  of  the  line  determined, 
one  has  to  find  the  perspective  of  but  one  other  point  by  the 
usual  method  (section  31)  and  then  join  these  two  perspectives 
by  an  indefinite  right  line. 

Example.  Consider  Fig.  58,  which  shows  an  indefinite  right 
line  A-B  together  with  its  projections  a-b  (horizontal)  and  a'-bf 
(vertical)  situated  in  the  second  quadrant,  and  a  point  of  sight, 
S,  together  with  its  projections  s  (horizontal)  and  s'  (vertical)  in 
the  first  quadrant.  The  line  thus  determined  is  found  (by  the 
usual  orthographic  method)  to  pierce  the  vertical  plane  at  PT\ 
this  point,  then,  according  to  the  above  is  a  point  in  the  perspective 
of  the  line.  (Note  that  the  correctness  of  the  point  Pf  is  established 
by  extending  the  line  A-B  on  through  the  planes  of  projection.) 

Having  one  point  in  the  perspective  of  the  line  thus  determined, 
select  any  second  point,  as  C  [do  not  forget  that  a  point  is  fixed 
by  its  two  projections,  c  (horizontal)  and  c'  (vertical)],  and  in 
accordance  with  section  31  find  its  perspective.  This  point  is 


PERSPECTIVE. 


66  ADVANCED  MECHANICAL  DRAWING 

found  to  be  C',  which  connected  with  Pf  by  a  right  line  determines 
the  perspective  of  the  indefinite  right  line  A-B. 

When  the  indefinite  right  line  is  parallel  to  the  picture  plane, 
two  points  in  the  line  must  be  selected  and  their  perspectives  found 
in  the  usual  way,  and  the  perspective  of  the  line  drawn  through 
the  perspective  of  the  points. 

36.  The  Vanishing-point  of  a  Line. — Lines  in  the  same  plane 
which  are  not  parallel  will  meet,  intersect;   parallel  lines  in  the 
same  plane  are  said  to  meet  at  infinity.     In  the  science  of  per- 
spective, when  two  parallel  lines  thus  meet  they  are  said  to  vanish. 
Assuming  two  parallel  lines,  then,  (i)   a  line  in  space  and  (2) 
a  line  parallel  to  it  through  the  point  of  sight,  the  lines  being 
no  exception  to  the  rule  will  meet — vanish — at  infinity;    now, 
assume  the  parallel  line  through  the  point  of  sight  to  be  the  line 
of  sight  connecting  this  meeting  or  vanishing-point  of  the  two 
lines  with  the  point  of  sight,  the  point  in  which  this  line  of  sight 
pierces   the  picture  plane  is  the  perspective  of   the   vanishing- 
point;   therefore,  the  vanishing-point  of   a   line  is  where  a  line 
parallel   to   it   through   the   point    0}    sight   pierces   the   picture 
plane. 

From  the  above  it  is  evident  that  a  system  of  parallel  lines 
has  a  common  vanishing-point,  for  the  line  through  the  point  of 
sight  parallel  to  one  line  is  parallel  to  all  of  them,  hence  the 
common  vanishing-point. 

37.  Rule  for  Finding  the  Vanishing-point  of  a  Line. — To  find 
the  vanishing-point  of  a  line  pass  a  parallel  line  through  the  point 
of  sight  and  find  where  it--the  parallel  line — pierces  the  picture 
plane;   that  is,  in  orthographic  projection,  through  the  horizontal 
projectioa  of  the  point  of  sight  draw  a  line  parallel  to  the  hori- 
zontal projection  of  the  given  line,  and  through  the  vertical  pro- 
jection of  the  point  of  sight  draw  a  line  parallel  to  the  vertical 
projection  of  the  given  line;   the  line  thus  determined  in  its  two 
projections  is  the  parallel  line  through  the  point  of  sight,  and 
the  point  in  which  it  pierces  the  picture  plane  is  the  vanishing- 
point  for  the  given  and  all  parallel  lines. 

Example.     In  Fig.  59  let  M-N  be  a  line  in  the  second  quad- 


PERSPECTIVE.  67 

rant,  and  let  5  be  a  point  of  sight  in  the  first  quadrant.  To  find 
the  vanishing-point  of  M-N,  through  the  horizontal  projection 
of  the  point  of  sight,  5,  draw  a  line  parallel  to  the  horizontal 


FIG.  59. 

projection  of  M-N,  m-n,  and  through  the  vertical  projection 
of  the  point  of  sight,  s',  draw  a  line  parallel  to  the  vertical  pro- 
jection of  the  given  line,  m'-n'\  the  point  V  in  which  the  line 


<58  ADVANCED  MECHANICAL  DRAWING. 

thus  shown  in  its  two  projections  pierces  the  vertical  or  picture 
plane  is  the  required  vanishing-point. 

38.  Rule   for  Finding  the  Perspective  of  a  Line. — Since  all 
lines  may  be  extended  and  then  considered  as  of  indefinite  length, 
by  combining  section  35  (which  shows  that  the  point  in  which 
a  line  pierces  the  picture  plane  is  a  point  in  the  perspective  of 
the  line)  with  section  36  (which  shows  that  the  vanishing-point 
of  a  line  is  a  point  in  the  perspective  of  the  line)  the  following 
definitive  rule  is  obtained: 

The  perspective  of  a  line  is  a  line  joining  the  point  in  which 
it  (the  line)  pierces  the  picture  plane  and  its  vanishing-point. 
(See  Fig.  59.) 

39.  The  Diagonal  and  Perpendicular. — There  are  two  special 
cases  of  the  right  line  much  used  in  perspective,  (i)  a  diagonal 
line,  and  (2)  a  perpendicular  line. 

The  diagonal.  A  diagonal  line  is  a  line  which  is  parallel 
with  the  horizontal  plane  of  projection  and  which  makes  an 
angle  of  forty-five  degrees  (45°)  with  the  vertical  plane. 

To  find  the  perspective  of  a  diagonal,  first  find  the  point  in 
which  it  pierces  the  picture  plane;  second,  find  its  vanishing- 
point,  and  third,  join  these  two  points  with  a  right  line;  this 
line  will  be  the  required  perspective. 

Fig.  60,  in  which  5  and  s'  are  the  horizontal  and  vertical  pro- 
jections, respectively,  of  the  point  of  sight,  and  m-n  and  m'-nf 
the  like  projections  of  a  diagonal,  illustrates  the  above  procedure. 
For  example,  to  find  the  vanishing-point  of  the  diagonal,  draw 
a  line  through  the  horizontal  projection  of  the  point  of  sight 
parallel  to  the  horizontal  projection  of  the  diagonal  (note  that 
this  line  makes  an  angle  of  45°  with  the  ground-line),  and  through 
the  'vertical  projection  of  the  point  of  sight  draw  a  line  parallel 
to  the  vertical  projection  of  the  diagonal  (note  that  this  line  is 
parallel  to  the  ground-line),  and  find  the  point  in  which  the  thus 
determined  line  pierces  the  vertical  plane;  this  point,  V,  will 
be  the  vanishing-point  of  the  diagonal. 

To  find  the  perspective  of  the  diagonal,  in  addition  to  the 
above,  find  the  point  Pf,  in  which  the  line  itself  pierces  the  vertical 


PERSPECTIVE. 


69 


FIG.  60. 


70  ADVANCED  MECHANICAL  DRAWING. 

plane,  then  join  it — the  point — with  the  vanishing-point  F — the 
line  will  be  the  required  perspective.     (Section  38.) 

The  perpendicular.  A  perpendicular  is  a  line  which  is 
parallel  to  the  horizontal  plane  of  projection  and  which  makes 
an  angle  of  ninety  degrees  (90°)  with  the  vertical  plane,  that  is, 
the  line  is  parallel  to  H  and  perpendicular  to  F. 

Referring  to  Fig.  61,  it  is  obvious  that  such  lines  vanish  at 
the  vertical  projection  of  the  point  of  sight  (section  36),  and,  since 
the  line  itself  is  seen  to  pierce  the  picture  plane  in  the  point  repre- 
senting the  vertical  projection  of  the  line,  the  perspective  of  the 
perpendicular  is  the  line  m'-nf-sf  joining  this  point  and  the 
vanishing-point  of  the  line.  (Section  38.) 

40.  Conventional  Method  for  Finding  Perspectives. — In  addi- 
tion to  the  elementary  method  already  given  for  finding  the 
perspective  of  a  point — that  of  finding  the  point  in  which  the 
line  of  sight  to  it  pierces  the  picture  plane — a  second  method 
is  available.  This  method  is  as  follows:  Pass  two  lines  through 
the  point,  then  find  the  perspectives  of  these  lines ;  the  intersection 
of  the  perspectives  will  be  the  perspective  of  the  point. 

The  two  lines  used  for  the  above  purpose  are  a  diagonal 
and  a  perpendicular.  These  two  lines  are  used  because  of  their 
convenience;  any  two  lines  may  be  used,  however. 

Example.  In  Fig.  62  let  5  and  sf  be  the  projections  of  the 
point  of  sight  and  p  and  /  the  projections  of  the  given  point, 
and  let  V  (found  as  in  section  36)  be  the  vanishing-point  for  all 
diagonals  inclined  to  the  left;  the  vanishing-point  for  all  perpen- 
diculars is  sf  (section  36).  To  find  the  perspective  of  the  point 
P  by  the  perpendicular-diagonal  method,  first  pass  a  perpen- 
dicular (P-O)  through  the  point.  This  is  found  to  pierce  the 
picture  plane  in  o'  (the  same  point  as  p',  which  represents  the 
vertical  projection  of  the  given  point),  and  by  section  38  its  per- 
spective is  found  to  be  the  line  o'-s*  joining  this  point  and  its 
vanishing-point;  next  pass  a  diagonal  line  (R-N)  through  the 
point,  find  the  point  n'  in  which  it  pierces  the  vertical  plane, 
then  join  this  point  with  the  vanishing-point  F';  this  line  (n'-V) 
will  be  the  perspective  of  the  diagonal.  The  intersection  of  the 


PERSPECTIVE. 


7i 


G 


Fio.  61. 


72  ADVANCED  MECHANICAL   DRAWING. 

perspective  of  the  perpendicular  and  the  perspective  of  the  di- 
agonal, P't  is  the  perspective  of  the  point. 


FIG.  62. 


Fig.  63  illustrates  the  finding  of  the  perspective  of  a  line  by 
the  perpendicular  diagonal  method,  two  points  in  the  line  being 
found  in  perspective,  then  joined  by  a  line. 


PERSPECTIVE. 


73 


This  seemingly  roundabout  method — the  perpendicular- diag- 
onal or    two-intersecting-lines   method — is  the  method   adopted 


FIG.  63. 

in  practical  perspective;    the  reasons  for  its  adoption  will  be- 
come apparent  as  the  discussion  progresses. 

41.  The  Horizon-line. — The  horizon  at  sea  is  that  bound- 
ing circle  of  vision  where  the  sea  seems  to  meet  the  sky;  on  land, 
barring  obstructions,  it  is  that  line  where  the  sky  and  earth 


74 


ADVANCED  MECHANICAL  DRAWING. 


seem  to  meet;  the  horizon  in  perspective  is  a  circle  the  plane  of 
which  is  parallel  to  the  horizontal  plane  of  projection  and  which 
passes  through  the  point  of  sight.  This  plane  is  of  infinite  extent, 


Horizon  Line 


FIG.  64. 

and,  it  is  obvious,  will  intersect  the  picture  plane  in  a  straight 
line  parallel  to  the  ground-line  passing  through  the  vertical 
projection  of  the  point  of  sight.  (See  Fig.  64.)  This  line  of 
intersection  is  called  the  horizon-line  of  the  picture. 

It  is  evident  that  all  lines  which  are  parallel  to  the  horizontal 
plane  will  vanish  at  some  point  in  the  horizon-line.     (Section  36.) 


PERSPECTIVE.  75 

42.  Distance  Points. — From  an  inspection  of  the  several 
figures  used  thus  far  to  illustrate  the  discussion  it  will  be  seen 
that  most  of  the  drawing  is  done  above  the  ground-line,  and  that 
the  horizontal  projection  of  the  point  of  sight  is  only  used  to  find 
the  vanishing-point  for  diagonals.  Since  the  horizontal  projection 
of  the  point  of  sight  is  so  little  used,  the  space  necessary  for  its 
depiction  is  saved  in  practice  by  omitting  it  entirely  and  assuming 
a  vanishing-point  for  diagonals  at  will;  this  point,  of  course,  is 
always  a  point  in  the  horizon-line. 

The  point  thus  assumed  may  be  either  to  the  fight  or  to  the 
left  of  the  vertical  projection  of  the  point  of  sight;  in  fact  it  is 
customary  to  assume  two  such  points — one  right  and  one  left— 
for  convenience  in  executing  the  perspective.  The  two  points 
thus  assumed  are  called  "  distance  points,"  and  fix  the  position 
of  the  horizontal  projection  of  the  point  of  sight,  as  witness  Fig. 
64:  Since  the  line  s-di  makes  an  angle  of  45°  with  the  ground- 
line,  and  since  the  horizon-line  is  parallel  to  the  ground-line, 
the  line  s'-di  =  0- di  =  0-5.  Hence  with  the  vertical  projection  of 
the  point  of  sight  and  the  distance  points  known,  to  find  the 
horizontal  projection  of  the  point  of  sight,  draw  an  indefinite 
vertical  line  through  the  vertical  projection  of  the  point  of  sight 
and  make  that  part  of  it  below  the  ground-line  equal  in  length 
to  the  distance  between  the  vertical  projection  of  the  point  of 
sight  and  either  distance  point;  the  lower  extreme  of  this  line  will 
be  the  required  horizontal  projection.  That  is,  the  horizontal 
projection  of  the  point  of  sight  is  as  far  in  front  of  the  vertical 
plane  as  the  distance  points  are  to  the  right  or  to  the  left  of  the 
vertical  projection  of  the  point  of  sight. 

To  illustrate  the  application  of  the  distance  points,  again 
consider  Fig.  64:  For  this  explanation  disregard  the  horizontal 
projection  of  the  point  of  sight,  and  let  it  be  required  to  find  the 
perspective  of  the  point  P  by  the  perpendicular-diagonal  method, 
and  let  the  two  projections  of  the  given  point  and  the  vertical 
projection  of  the  point  of  sight  be  the  known  conditions.  To 
find  the  perspective  of  the  point,  first  draw  the  horizon-line  and 
assume  the  distance  points,  then  (using  the  distance  point  on 


76  ADVANCED  MECHANICAL  DRAWING. 

the  left)  find  the  perspective  of  the  point  as  usual  (section  40); 
this  is  seen  to  be  Pf . 

Suppose  one  finds  it  more  convenient  to  use  a  diagonal  which 
is  inclined  to  the  right  instead  of  one  inclined  to  the  left  as  in  the 
above  case,  by  definition  (section  36)  such  diagonals  are  found  to 
vanish  at  d '  on  the  right,  and  carrying  the  perspective  through 
with  such  a  diagonal  line  the  perspective  of  the  point  is  found 
to  check  with  that  found  by  using  a  diagonal  of  opposite  inclination. 

From  the  above,  then,  it  is  evident  that  it  is  optional  which 
distance  point  is  used. 

43.  The  Perspective  of  a  Plane  Figure.— In  Fig.  65  let  1-2-3-4-5 
represent  the  horizontal  projection  of  a  five-sided  polygon  and 
let  i'-2'-3'-4'-5'  represent  its  vertical  projection,  and  let  s'  be 
the  point  of  sight,  X-Y  the  horizon-line,  and  df  and  dr  the  distance 
points. 

To  find  the  perspective  of  the  figure,  find  the  perspective, 
separately,  of  each  of  the  five  corner  points,  using  the  perpen- 
dicular-diagonal method,  and  join  these  five  perspectives  in 
the  same  order  as  they  occur  in  the  original  with  right  lines, 
i.e.,  point  i  to  point  2,  2  to  3,  etc.;  the  resulting  figure  will  be 
the  required  perspective. 

For  example,  consider  point  i:  the  perpendicular  through 
it  is  shown  in  perspective  as  the  line  i'-s',  the  diagonal  through 
it  is  shown  in  perspective  as  the  line  o'-ft ',  and  the  perspective 
of  the  point  as  the  point  i',  the  intersection  of  the  two  above 
perspectives;  the  perspective  of  point  5  is  found  in  a  similar 
manner  and  is  seen  to  be  point  5';  the  perspective  of  the  line 
1-5,  then,  is  clearly  the  line  i'-5'  joining  the  points  i'  and  5'. 

It  will  be  remarked  that  but  one  of  the  distance  points  has 
been  used  in  the  figure;  the  perspective  may  be  found  by  using 
either  one  or  both. 

44.  Special    Cases    of   the    Right  Line. — Before   proceeding 
further  with  the  discussion,  it  is  desired  to  call  the  attention  of 
the  student  to  a  number  of  special  cases  of  the  right  line,  as  these 
well  fixed  in  mind  enable  one  to  recognize  the  accuracy  of  a 
perspective  at  a  glance,  and   in  the   execution  of   a  drawing  to 


PERSPECTIVE. 


77 


7»  ADVANCED  MECHANICAL  DRAWING. 

greatly  expedite  the  labor  of  construction.    These  "helps"  or 
"aids"  are  as  follows: 

1.  Every  system  of  parallel  lines,  unless  parallel  to  the  picture 
plane,  has  a  common  vanishing-point.     (Section  36.) 

2.  The  perspective  of  a  line  which  is  parallel  to  the  picture 
plane  is  parallel  to  the  line  itself.     (See  Fig.  66.)     If  the  line 
is  parallel  to  both  planes  of  projection — a  horizontal  line  parallel 
with  the  picture  plane — its  perspective  is  a  line  which  is  parallel 


Horizon 


Line 


d' 


Ground 


Line 


FIG.  66. 


with  the  ground-line,  as  witness  the  line  1-5  in  Fig.  65.  If  the 
line  is  perpendicular  to  the  horizontal  plane  and  parallel  to  the 
picture  plane,  its  perspective  is  a  line  which  is  perpendicular  to 
the  ground-line.  (See  Fig.  67.) 

3.  The  perspective  of  a  line  or  point  which  is  in  the  picture 
plane  is  the  line  or  point  itself;  that  is,  such  a  perspective  would 
be  "full-size." 

45.  The  Plan  and  Elevation  Moved  from  the  Field  of  the 
Picture. — By  referring  to  Fig.  65,  which  shows  the  plan  and 
elevation  in  projection  and  in  their  assumed  position  with  reference 
to  the  picture  plane,  one  remarks  the  apparent  confusion  of  lines, 
and  can  readily  imagine  the  real  confusion  which  would  occur 
in  a  more  complex  perspective,  as  the  perspective  of  a  house 
if  it  were  drawn  in  a  similar  manner.  To  minimize  this  con- 


PERSPECTIVE. 


79 


fusion  of  lines  the  plan  and  elevation  are  separated — no  longer 
in  projection — and  are  removed  from  the  field  of  the  picture 
and  arranged  and  used  as  follows: 

Let  Fig.  68,  A,  represent  the  conditions  for  a  certain  example 
in  perspective,  p  and  //  being  the  projections  of  a  point  in  the 
second  quadrant,  A  distance  above  the  horizontal  plane  and  B 
distance  away  from  the  vertical  plane,  and  s  and  sf  the  projections 


Line 


O 
FIG.  67. 

of  a  point  of  sight,  C  distance  in  front  of  the  vertical  or  picture 
plane,  and  D  distance  above  the  horizontal  plane,  and  E  distance 
to  the  right  of  the  assumed  point. 

To  find  the  perspective  of  the  point  under  the  above  con- 
ditions, and  at  the  same  time  to  keep  the  projections  of  the  point 
out  of  the  field  of  the  picture,  it  is  necessary  to  employ  an  auxiliary 
ground-line,  Fig.  68,  B,  delineating  the  usual  arrangement.  To 
execute  this  figure,  with  the  field  of  the  picture  (that  portion  of 
the  drawing  surface  to  receive  the  picture)  known,  draw  the 
ground-line  and  locate  the  point  s*  (the  vertical  projection  of  the 


or  TMC 
UNIVERSITY 


8o 


ADVANCED  MECHANICAL   DRAWING. 


point  of  sight),  D  distance  above  it;  through  sf  draw  the  horizon- 
line,  and  on  it  locate  the  distance  point  df  [note  that  this  is  C 
distance  to  the  left  (may  be  either  right  or  left)  of  sf  and  repre- 
sents or  corresponds  to  the  distance  the  point  of  sight  is  in  front  of 
or  away  from  the  picture  plane];  next  draw  an  auxiliary  ground- 
line  at  any  convenient  point  above  the  field  of  the  picture  and 
locate  the  horizontal  projection  or  plan  of  the  point  B  distance 


s' 


If 

sit! 


CO 

•*V 

1 

*\ 

Auxiliary 

Ground 

Lm« 

<  

E  > 

/ 

Y 

V 

f 


Plorizon                      Line 
c , 


Ground  Line 


S 

FIG.  68,  A. 


FIG.  68,  B. 


away  from  it,  and  E  distance  to  the  left  of  the  location  of  the 
point  of  sight  (note  that  the  dimensions  B  and  E  correspond  to 
the  dimensions  B  and  E  of  Fig.  68,  .4);  lastly,  locate  the  vertical 
projection  or  elevation  of  the  given  point  p*  with  reference  to 
the  ground-line  (A  distance  above  it),  and  at  some  convenient 
point  to  the  left  (may  be  right  or  left)  of  the  field  of  the  picture. 

That  is,  the  plan  or  H  projection  is  located  with  reference 
to  one  ground-line,  and  the  elevation  or  V  projection  with  refer- 
ence to  a  second  ground-line,  and  while  the  principle  of  con- 
ventional projection  is  violated  (since  the  two  projections  do 
not  lie  in  a  perpendicular  to  a  common  ground-line)  the  location 
of  the  point  is,  nevertheless,  absolutely  and  accurately  fixed. 

To  find  the  perspective  of  the  point  under  this  new  arrange- 
ment, first  draw  an  elevation  line  X-Y  across  the  field  of  the 


PERSPECTIVE.  8 1 

picture  (obtained  by  drawing  a  horizontal  line  through  the  ele- 
vation of  the  point,  said  to  be  "  projected  in  ")  representing  the 
elevation  or  distance  above  the  ground-line  of  the  given  point, 
then  pass  a  perpendicular  (section  39)  through  the  point;  this 
pierces  the  picture  plane  at  X,  and  is  shown  in  perspective  as  the 
line  X-s'  (section  38) ;  next  pass  a  diagonal  (section  39)  through 
the  point;  this  pierces  the  picture  plane  at  F,  and  is  shown  in 
perspective  as  the  line  Y-d' ;  the  intersection  of  these  two  per- 
spectives, P'j  is  the  required  perspective.  (Section  40.) 

Fig.  69,  A,  depicts  the  conditions  for  a  second  example  in 
perspective,  and  shows  the  projections  of  a  line  M-N  and  a  point 
of  sight  S.  Fig.  69,  B,  shows  the  conventional  arrangement 
of  the  above  conditions  and  delineates  the  finding  of  the  per- 
spective of  the  line. 

46.  Practical  Perspective. — The  arrangement  of  the  plan 
and  elevation — the  projections — of  an  object,  and  of  the  point 
of  sight,  the  two  ground-lines,  etc.,  shown  in  Figs.  68,  B,  and  69, 
B,  is  the  one  adopted  for  practical  purposes.  The  statement 
has  been  made  that  the  plan  and  elevation  are  removed  from 
the  field  of  the  picture  to  minimize  the  confusion  of  the  lines; 
in  the  two  examples  just  discussed,  it  will  be  remarked  that  this 
new  arrangement  of  the  plan  and  elevation  has  not  simplified 
matters  at  all;  rather,  since  one  has  to  project  down  from  the 
plan  and  in  from  the  elevation,  it  has  added  new  lines  to  the 
drawing  and  increased  the  confusion  of  lines.  This  is  true  in  the 
examples  given,  and  if  all  examples  were  as  simple  as  these,  it 
would  be  better  to  use  the  method  described  in  section  40,  wherein 
the  plan  and  elevation  are  in  projection  and  occupy  the  field 
of  the  picture.  In  practice,  however,  one  deals  with  many  points 
and  lines  in  a  single  object,  and  with  the  plan  and  elevation  out 
of  the  field  of  the  picture,  the  points  of  the  object  are  projected 
into  it  by  means  of  the  T  square  and  triangles  and  the  perspective 
of  lines  obtained  by  many  " short  cuts";  thus  one  finds  it  neces- 
sary to  draw  but  few  lines,  and  the  field  of  the  picture  is  kept 
comparatively  free  from  confusion. 

Figs.  68,  B,  and  69,  B,  delineate  all  that  is  fundamental  in 


82 


ADVANCED  MECHANICAL  DRAWING. 


practical  perspective,  and  should  be  carefully  studied  and  thor- 
oughly understood,  as  the  examples  discussed  from  this  point  on 


cannot  be  understood  without  a  first  understanding  of  the  above. 

As  a  first  real  example  in  practical  perspective — the  above 

examples   being  very  elementary — let  Fig.  70,  A,  be  the  projec- 


PERSPECTIVE. 


tions  of  a  small  oilstone  mounted  in  a  wooden  box,  and  let  it 
be  required  to  picture  it  as  resting  on  the  horizontal  plane  with 


-**=: 

[*—  -,\  —  > 

ft 

__§_ 

0. 

_ 

o 

I 


its  long  sides  parallel  with  the  vertical  plane  and  its  nearest 
side  ft"  away  from  it. 


84  ADVANCED  MECHANICAL  DRAWING. 

Fig.  70,  B,  shows  the  disposition  of  the  plan  and  elevation 
in  accordance  with  the  previous  explanation,  the  assumption 
of  a  viewpoint,  s',  the  horizon-line,  and  a  distance  point,  df. 

These  preliminaries    arranged,   the    perspective  is  found  as 
follows : 

Through  point  i  pass  a  perpendicular;  this  is  seen  to  pierce 
the  vertical  plane  at  Y  (Y  being  the  vertical  projection  of 
point  i  obtained  by  projecting  in  from  the  elevation  at  the  side 
parallel  with  the  ground-line  to  an  intersection  with  the  perpen- 
dicular from  the  plan),  and  since  all  perpendiculars  vanish  at 
5',  its  perspective  is  the  line  Y-s'\  next  pass  a  diagonal  through 
the  point  i  (through  the  horizontal  projection — plan — draw  a 
line  at  45°  with  the  ground-line,  and  through  the  vertical  pro- 
jection— elevation — draw  a  line  parallel  with  the  ground-line); 
this  is  seen  to  pierce  the  picture  plane  at  X,  and  since  it  vanishes 
at  d!  its  perspective  is  the  line  X-d'\  the  intersection  of  the  two 
perspectives — the  perpendicular  and  diagonal — is  the  required 
perspective;  this  is  seen  to  be  point  i'.  Proceeding  in  this 
manner  the  perspective  of  all  of  the  corner  points  may  be  obtained 
— one  at  a  time — and  when  properly  joined  by  right  lines  will 
give  the  required  perspective. 

The  labor  of  construction  can  be  reduced  by  applying  section 
44.  For  example,  having  found  the  perspective,  i',  of  the  corner 
point  marked  i,  the  perspective  of  the  line  1-2  may  be  found 
as  follows:  The  line  is  known  to  be  parallel  with  both  planes  of 
projection,  and  hence,  that  its  perspective  will  be  parallel  to  the 
ground-line,  therefore,  since  one  point  and  direction  will  determine 
a  line,  draw  a  horizontal  line  through  point  i'  and  terminate  it 
where  it  crosses  the  perspective  of  the  perpendicular  through 
point  2.  By  this  procedure  not  only  is  point  2  found  in  per- 
spective but  the  perspective  of  the  line  1-2  is  obtained  at  the  same 
time,  and  this,  too,  without  the  use  of  both  a  perpendicular 
and  diagonal.  (The  above  mentions  the  use  of  a  perpendicular 
to  terminate  the  perspective  of  the  line  1-2;  either  a  perpendicular 
or  a  diagonal  may  be  used — it  is  not  necessary  to  employ  both — 
that  one  being  used  which  will  give  the  sharper  intersection; 


PERSPECTIVE.  85 

that  is,  if  the  given  perspective  and  the  perspective  of  the  per- 
pendicular are  so  nearly  parallel  that  it  is  difficult  to  determine 
the  exact  intersection  of  the  two,  it  is  well  to  use  a  diagonal,  as 
it  is  probable  that  it  will  give  a  sharp  intersection — will  cross  the 
other  line  more  nearly  at  right  angles.) 

The  student  should  analyze  the  disposition  made  of  the  plan 
and  elevation  in  the  foregoing  explanation  and  see  for  himself 
that  it  is  nothing  new,  and  that  all  of  the  principles  of  ortho- 
graphic projection  are  observed  in  that  a  second  vertical  pro- 
jection— elevation — is  obtained,  a  point  at  a  time,  which  is  in 
projection  with  the  plan  by  projecting,  horizontally,  in  from 
the  elevation  at  the  side  as  needed  to  a  point  in  projection, 
vertically,  with  the  same  point  of  the  plan.  Furthermore,  that 
the  auxiliary  ground-line  used  gives  the  same  results  as  would 
the  use  of  the  real  ground-line. 

47.  Parallel   Perspective. — When    an    object    is    so    situated 
that  a  large  number  of  its  principal  lines  are  parallel  to  the  picture 
plane,  the  perspective  obtained  is  said  to  be  a  "parallel"  per- 
spective*   Fig.  70,  By  is  an  example  of  parallel  perspective. 

Parallel  perspective  is  very  simple  and  convenient,  involving 
the  use  of  only  the  point  of  sight  and  distance  points,  and  since 
most  of  the  lines  of  the  object  are  either  parallel  or  perpendicu- 
lar to  the  picture  plane,  after  one  or  two  principal  points  are 
found  by  the  perpendicular-diagonal  method  the  perspective  can 
be  quickly  finished  by  the  "  one-point-and-direction "  method. 

In  this  class  of  perspective  it  is  clearly  evident  that  the  per- 
spective of  a  circle  the  plane  of  which  is  parallel  to  the  picture 
plane  will  be  a  true  circle;  the  perspective  of  a  hexagon,  a  square, 
etc.,  similarly  situated  will  be  true  figures — points  which  facilitate 
the  drawing. 

48.  Oblique    or  Angular    Perspective. — If   an    object   is   so 
situated  relative  to  the  picture  plane  that  a  large  number  of  its 
principal  lines  are  not  parallel  to  the  plane  but  make  a  known 
angle  with  it,  the  perspective  obtained  is  said  to  be  an  "angular 
perspective";  Fig.  71  is  an  example  of  this  class. 

Angular  perspective  involves  the  use  of  a  point  of  sight, 


86  ADVANCED  MECHANICAL  DRAWING. 

distance-points,  and  vanishing-points  for  all  systems  of  parallel 
lines.  In  Fig.  71  it  will  be  seen  that  all  of  the  lines  of  the  object 
are  either  at  60°  or  30°  with  the  picture  plane  and  require  two 
vanishing-points.  These  are  found  as  follows: 

The  ground-line  and  the  point  of  sight  assumed,  draw  an 
indefinite  perpendicular,  M-M,  through  the  vertical  projection  of 
the  point  of  sight  s',  and  draw  the  horizon-line;  next,  assume  one 
of  the  vanishing-points,  in  this  case  say  that  for  the  30°  lines,  the 
point  F',  and  from  this  vanishing-point  drop  a  perpendicular  Y'-Y 
to  the  ground-line,  and  from  the  point  of  intersection,  F,  draw  a  line 
making  an  angle  of  30°  with  the  ground-line  and  inclined  opposite 
in  direction  to  the  inclination  of  the  30°  lines  of  the  object,  until 
it  intersects  the  perpendicular  M-M  through  the  point  of  sight; 
from  this  point,  s",  draw  a  60°  line,  s"-X,  opposite  in  direction  to 
the  60°  lines  of  the  object,  to  the  ground-line,  and  from  the  point  X 
in  which  it  intersects  the  ground-line  erect  a  perpendicular,  X-X' , 
to  an  intersection  with  the  horizon-line;  this  point  (point  X')  will 
be  the  vanishing-point  for  all  60°  lines. 

An  analysis  of  the  above  will  show  that  it  is  the  same  as  the 
procedure  given  in  section  37,  if  one  will  but  consider  the  hori- 
zontal projection  of  the  point  of  sight  as  having  been  revolved, 
for  convenience,  over  into  the  second  quadrant. 

To  find  the  perspective  in  this  case,  use  a  perpendicular  and 
either  a  30°  or  60°  line  method.  That  is,  to  find  the  per- 
spective of  point  i,  for  example,  pass  a  perpendicular  H-K 
through  the  point  and  find  where  it  pierces  the  picture  plane 
(point  K')  by  projecting  in  from  the  elevation,  and  then  draw 
its  perspective  (K'-s1)  by  joining  this  point  with  s',  the  vanishing- 
point  for  perpendiculars;  next,  in  place  of  passing  a  45°  diagonal 
through  the  point,  pass  either  a  30°  or  60°  diagonal  line  [a  line 
at  30°  or  60°  with  the  picture  plane  and  parallel  to  the  horizontal 
plane  (a  special  diagonal),  as  the  line  C-D  or  E-F],  find  where  it 
pierces  the  picture  plane  (point  F'  or  Z>'),  and  draw  its  perspective 
(F'-Y'  or  D'-X')  by  joining  this  point  with  its  vanishing-point. 
The  intersection  of  this  perspective  with  the  perspective  of  the 
perpendicular  (K'-sf)  is  the  required  perspective. 


PERSPECTIVE. 


ADVANCED  MECHANICAL  DRAWING. 

The  same  point  can  be  found  by  using  the  45°  diagonal,  as 
shown  by  the  drawing,  or  in  fact  any  degree  diagonal  provided 
the  proper  vanishing-point  is  used,  the  advantage  in  this  par- 
ticular case  being  that  the  use  of  either  the  30°  or  60°  diagonal 
not  only  gives  the  point  desired,  but  at  the  same  time  gives  the 
lines  of  the  object. 

49.  How  to  Assume  Conditions. — As  has  been  explained 
in  the  early  part  of  these  remarks,  it  is  customary  to  assume  the 
object  as  situated  in  the  second  quadrant,  and  at  the  same  time 
it  was  shown  that  the  nearer  the  object  was  to  the  picture  plane 
the  larger  its  perspective  would  be,  and  that  the  farther  away 
from  the  picture  plane,  the  smaller  the  perspective  of  the  object 
would  become.  It  is  obvious,  then,  that  the  position  of  the 
plan  with  reference  to  the  auxiliary  ground  determines  the  size 
of  the  perspective. 

For  obvious  reasons  most  objects  are  assumed  to  rest  on  the 
horizontal  plane.  This,  of  course,  places  the  elevation  on  the 
ground-line. 

The  point  of  sight  is  taken  with  reference  to  those  faces  of 
the  object  it  is  desired  to  depict.  That  is,  if  the  top  face  is  desired 
in  connection  with  certain  side  faces,  the  point  of  sight  must 
obviously  be  above  the  elevation  of  the  top  face — made  manifest 
on  the  drawing  by  the  elevation  of  the  vertical  projection  of  the 
point  of  sight  referred  to  the  elevation  of  the  top  of  the  elevation 
of  the  object.  The  side  faces  desired  are  made  manifest  on  the 
drawing  by  the  disposition  of  the  plan  and  the  horizontal  projection 
of  the  point  of  sight;  the  sides  of  the  plan  which  are  nearest 
the  picture  plane  are  the  ones  that  will  be  seen,  and  how  much  of 
each  is  determined  by  the  position  of  the  horizontal  projection  of 
the  point  of  sight — whether  to  one  side  or  not,  and  its  distance 
from  the  picture  plane. 

For  economy  of  space  it  is  well  to  assume  the  first  quadrant 
horizontal  projection  of  the  point  of  sight  to  be  revolved  over 
into  the  horizontal  plane,  second  quadrant,  as  shown  in  Fig.  71, 
and  when  so  assumed  will  reverse  the  direction  of  inclination 


PERSPECTIVE.  89 

of  the  H  projection  of  any  lines  inclined  to  the  picture  plane 
passing  through  the  point  of  sight. 

It  is  obvious  that  ihe  farther  away  the  point  of  sight  is  from 
the  picture  plane,  the  greater  the  distance  between  the  vertical 
projection  of  the  point  of  sight  and  any  vanishing-points  used, 
and,  of  a  consequence,  the  slower  convergence  of  the  lines  of 
the  picture.  From  this,  then,  it  is  seen  that  to  keep  the  conver- 
gence from  becoming  conspicuous,  the  vanishing-points  should  be 
as  "wide"  as  possible,  in  which  case  it  is  well  to  apply  the  fore- 
going principles  as  follows: 

In  Fig.  71,  when  two  vanishing-points  are  required,  let  the 
line  X'-X  represent  one  horizontal  extreme  of  the  available 
drawing-surface,  say  the  left-hand  side  or  end  of  the  drawing- 
board,  and  let  Y'-Y  represent  the  other  horizontal  extreme,  as 
the  right  end  of  the  drawing-board,  and  let  the  horizontal  line 
X-Y  represent  the  lower  vertical  extreme  of  the  drawing-surface, 
say  the  bottom  of  the  drawing-board. 

From  the  lower  left-hand  corner,  AT,  draw  the  60°  line  X-s", 
and  from  the  lower  right-hand  corner  draw  the  30°  line  F-s", 
and  produce  these  lines  to  an  intersection  s";  this  point  will 
be  the  revolved  position  of  the  horizontal  projection  of  the  point 
of  sight  which  will  give  the  "widest"  vanishing-points  possible 
on  the  drawing-surface  used. 

The  projections  of  the  point  of  sight  assumed,  it  is  then  an 
easy  matter  to  arrange  the  plan  and  elevation,  to  give  the  desired 
perspective  or  view,  and  to  locate  all  vanishing-points,  the  horizon- 
line,  etc.  That  is,  it  is  sometimes  convenient  to  assume  the 
conditions  "  backwards, "  so  to  speak,  to  fit  the  drawing-board. 

In  cases  where  the  elevation  of  a  point  coincides  with  the 
elevation  of  the  point  of  sight,  or  in  cases  where  it  is  difficult  to 
obtain  a  sharp  intersection  of  the  perpendicular  and  diagonal 
used  because  of  their  inclination,  as  the  lines  A'-s*  and  B'-d' ', 
Fig.  72,  it  is  well  to  assume  an  auxiliary  elevation  for  the  point, 
then  find  its  perspective  at  this  elevation  and  project  it  (vertically) 
onto  either  the  perpendicular  or  diagonal  drawn  at  the  correct 
elevation. 


90  ADVANCED  MECHANICAL  DRAWING. 

50.  The  Perspective  of  Shadows. — Before  attempting  to 
digest  this  section  the  student  should  have  a  working  knowledge 
of  the  principles  of  cast  shadows — such  as  may  be  derived  from 
a  perusal  of  Chapter  II;  this  knowledge,  together  with  an  under- 
standing of  the  elementary  principles  of  perspective,  renders 
the  finding  of  the  perspective  of  the  shadow  cast  by  an  object  a 
very  simple  procedure. 


Elevation 


Line 


Ground  Line 


FlG.  72. 

Shadows  are  rarely  shown  in  their  true  outline  in  practical 
draughting,  but  are  approximated  in  accordance  with  the  draughts- 
man's conception  of  them;  that  is,  from  his  experience  and 
observation  he  is  able  to  "guess"  in  a  shadow  which  is  approx- 
imately true  to  life.  For  this  reason,  and  since  the  art  is  seldom 
applied  by  the  usual  engineer-draughtsman,  it  is  proposed  to 
treat  the  subject  very  briefly,  explaining  the  fundamentals  and 
giving  a  few  elementary  examples  only. 


PERSPECTIVE.  91 

Theory.  The  theory  is  the  same  as  that  for  all  shadows, 
i.e.,  every  surface  is  uniformly  covered  with  light  except  in  so 
far  as  the  rays  of  light  are  intercepted  by  a  point,  line,  or  object, 
and  of  a  consequence  leaving  a  portion  of  the  surface  unilluminated 
representing  the  shadow  of  the  point,  line,  or  object. 

The  theory  is  applied  exactly  as  in  the  orthographic  projection 
of  shadows. 

Application  of  the  theory.  To  find  the  perspective  of  the 
shadow  cast  by  the  perspective  of  a  point,  pass  the  perspective 
of  a  ray  of  light  through  the  perspective  of  the  point  and  find 
the  point  in  which  the  perspective  of  the  ray  pierces  the  per- 
spective of  the  planes  of  projection. 

Now,  the  object  (point  in  this  case)  being  always  assumed 
in  the  second  quadrant,  and  the  light  conventionally  assumed  as 
making  an  angle  of  45°  with  the  planes  of  projection  (first  quad- 
rant) as  shown  in  Fig.  73,  it  is  evident  that  the  shadow  always 
falls  on  the  horizontal  plane,  except,  of  course,  that  part  of  it 
which  falls  on  the  object  itself.  The  finding  of  the  perspective 
of  the  shadow  cast  on  the  horizontal  plane  is  the  principal  point 
in  the  perspective  of  shadows,  as  in  most  objects  there  are  planes 
which  are  parallel  with  the  horizontal  plane,  and  with  the  shadow 
located  on  these  and  on  H,  with  a  knowledge  of  shadows  (such 
as  the  shadow  of  a  line  on  a  parallel  plane  is  parallel  and  equal 
to  the  line,  etc.)  the  shadow  on  vertical  and  other  planes  is  easily 
located. 

As  in  all  perspective  work  where  there  are  parallel  lines  it  is 
necessary  to  locate  a  vanishing-point  for  the  lines,  it  is  necessary 
to  locate  a  vanishing-point  for  the  rays  of  light  and  for  the  hori- 
zontal projection  of  rays  In  Fig.  73,  then,  which  represents 
the  perspective  of  a  small  rectangular  block,  let  s  and  sf  be  the 
projections  of  the  point  of  sight,  let  the  light  be  at  45°  to  the 
planes  of  projection,  and  note  how  these  vanishing-points  are 
obtained. 

Through  the  vertical  projection  of  the  point  of  sight,  5/,  draw 
the  line  S'-TI  parallel  to  the  vertical  projection  of  the  rays  of  light, 
and  through  the  horizontal  projection  of  the  point  of  sight,  $, 


92 


ADVANCED  MECHANICAL  DRAWING. 


draw  the  line  s-r  parallel  with  the  horizontal  projection  of  the 
rays ;  the  point  r\  in  which  this  line  pierces  the  vertical  or  picture 


plane,  is  the  vanishing-point  for  all  rays  of  light.  (Section  37.) 
In  a  similar  manner  the  point  V*  is  found  to  be  the  vanishing- 
point  for  all  horizontal  projections  of  rays. 


PERSPECTIVE.  93 

To  apply  these  two  vanishing-points  to  find  the  shadow  of 
a  point,  say  point  3,  Fig.  73,  proceed  as  follows: 

It  is  evident  that  the  point  3  itself  is  one  point  in  the  per- 
spective of  the  ray  of  light  through  the  point,  and  since  all  rays 
vanish  at  r\ ,  the  line  $-r\  is  the  perspective  of  a  ray ;  in  like  manner, 
it  is  evident  that  the  point  4  is  the  horizontal  projection  of  the 
point  3  (the  object  being  assumed  to  rest  on  H,  and  the  line — 
edge — 3-4  being  perpendicular  to  //),  and  being  the  horizontal 
projection  is  one  point  in  the  perspective  of  the  horizontal  pro- 
jection of  the  ray  of  light  through  point  3,  and  since  all  horizontal 
projections  of  rays  vanish  at  r7,  the  line  4-r/  is  the  perspective 
of  the  horizontal  projection  of  the  ray.  Now,  it  is  evident  that 
the  shadow  of  the  point  must  necessarily  lie  in  the  perspective 
of  the  ray,  and  also  in  the  perspective  of  the  H  projection  of 
the  ray,  therefore,  the  shadow  of  the  point  is  (3')  the  intersection 
of  these  two  lines. 

Examples.  After  selecting  the  points  and  lines  which  cast 
the  outline  of  the  shadow,  a  series  of  shadow- points  are  found 
in  the  above  manner,  and  these  properly  joined  together  give 
the  desired  shadow,  as  witness  the  points  3'~7'-6',  etc.,  Fig.  73. 
The  shadow  on  the  H  plane  drawn,  it  is  an  easy  matter  to  determine 
those  faces  of  the  object  which  are  in  the  shadow,  as  the  face 

4-3-7-8- 

Fig.  74  represents  the  perspective  of  a  carriage-block.  In 
this  example  note  the  position  of  the  horizontal  projection  of  the 
point  of  sight,  s  (revolved  into  the  second  quadrant  for  con- 
venience), and  the  construction  used  in  locating  the  vanishing- 
points  TI  and  r1 .  The  figure  shows  the  perspective  of  the  shadow 
on  the  H  plane,  showing  the  work,  point  by  point,  and  also  the 
shadow  cast  on  the  object  itself.  (In  the  shadow  on  the  H  plane 
note  the  point  9' ;  how  the  point  9  is  projected  onto  H  at  X,  that 
the  perspective  of  the  H  projection  of  the  ray  through  point  9 
may  be  drawn.) 

The  figure  introduces  the  finding  of  the  perspective  of  the 
shadow  on  a  plane  other  than  the  H  plane.  The  plane  in  this 


94 


ADVANCED  MECHANICAL  DRAWING. 


PERSPECTIVE. 


95 


LLJ     -,- 

35 


o  o 


'••:, 


"*\  ,  <* 


48- 


96  ADVANCED  MECHANICAL  DRAWING. 

case  being  a  plane  parallel  with  H  the  procedure  is  very  simple 
and  is  as  follows: 

Through  the  point  3  draw  the  line  3-^1,  the  perspective  of  a 
ray,  and  through  the  point  4,  the  H  projection  of  point  3 
on  the  plane  of  the  shadow,  draw  the  line  4-^,  the  perspective 
of  the  H  projection  of  the  ray  on  the  new  plane;  the  intersection 
of  these  two  lines,  point  3',  is  the  shadow  of  point  3,  and  since 
the  line  3-9  is  parallel  to  the  plane  its  shadow  on  the  plane  will 
be  parallel  to  itself.  The  shadow  is  finished  by  drawing  a  line 
through  the  point  3'  parallel  to  the  line  3-9,  i.e.,  to  the  same 
vanishing-point. 

The  foregoing  examples,  while  being  specific,  are  illustrative 
of  the  method  for  finding  the  shadow  of  a  point  on  any  plane, 
the  rule  for  which  is  as  follows: 

To  find  the  perspective  of  the  shadow  oj  a  point  on  any  planey 
through  the  perspective  oj  the  point  draw  the  perspective  oj  a  ray, 
and  through  the  perspective  oj  the  projection  oj  the  point  on  the 
plane  draw  the  perspective  oj  the  projection  (on  this  plane)  oj 
the  ray;  the  intersection  oj  these  two  lines  will  be  the  required 
shadow. 

Remarks.  In  all  shadow  work,  orthographic  or  sceno- 
graphic  (perspective)  projection,  it  is  necessary  that  the  draughts- 
man be  able  to  read  his  drawing  well,  and  thus  be  able  to  select 
those  lines  and  points  which  cast  the  outline  of  the  shadow; 
with  this  ability  and  a  knowledge  of  the  principles  of  cast  shadows 
and  perspective  many  short  cuts  are  open  to  the  draughtsman 
whereby  the  execution  of  the  work  is  greatly  expedited.  This 
skill  of  execution  can  only  be  acquired  by  practice,  and  it  is 
recommended  that  the  student  analyze  all  of  his  constructions  and 
see  wherein  they  might  have  been  shortened.  Such  a  practice 
in  a  comparatively  few  examples  will  suffice  to  give  a  knowledge 
of  the  shortest  construction  to  use. 

In  all  of  the  examples  discussed  the  light  was  assumed  to  make 
an  angle  of  45°  with  the  planes  of  projection.  This  assumption 
is  simply  a  conventional  one;  the  light  may  be  assumed  at  will, 
as  witness  Fig.  75. 


PART   II. 

EXERCISES. 


CHAPTER  IV. 
THEORETICAL  PROBLEMS. 

51.  Explanatory. — The  following  problems  are  given  as  ex- 
ercises in  drawing  calculated  to  perfect  a  working  knowledge 
of  the  principles  of  Descriptive  Geometry,  and,   together  with 
the  problems  given  in  Chapter  V,  to  form  a  series  of  exercises 
for  a  course  in  advanced  drawing. 

The  work  is  so  designed  that  the  executed  solution  of  one  or 
more  examples  constitutes  an  exercise,  and  the  exercises  so 
finished  and  lettered  as  to  form  a  drawing  sheet  or  plate. 

52.  General  Directions. — The  paper  used  for  the  course  should 
be  a  good  quality  of   drawing-paper,  and    should   be  9"Xi2", 
or  a  little  greater,  in  dimensions;   the  border-line  will  be  a  rect- 
angle 8"Xn"  in  dimensions,  and  the  finished  sheet  should  have 
a  J"  margin  on  all  four  sides  outside  of  this.     (See  Fig.  76.) 

The  notation  used  to  be  that  shown  in  Fig.  76,  and  to  be 
used  in  strict  accordance  with  the  conventions  of  Descriptive 
Geometry,  i.e.,  a  vertical  projection  of  the  second  quadrant  is 
a  hidden  line,  etc. 

The  conditions  for  the  problems  are  to  be  those  given,  when 
such  is  the  case,  otherwise,  they  are  to  be  assumed  at  the  draughts- 

97 


98 


ADVANCED  MECHANICAL  DRAWING. 


man's  discretion.     (Always  assume  a  figure  of  sufficient  size  to 
afford  a  clean-cut,  legible  solution.) 

All  lettering  to  be  free-hand,  single-line  Gothic  letters;  the 
letters  may  be  upright  or  inclined.  The  title  letters  to  be  capitals 
or  upper-case  letters,  initial  letters  TV'  high,  other  letters  J" 
high.  The  name  and  date,  and  all  other  lettering,  to  be  of  the 
lower-case  or  small  letters,  initial  letters  excepted,  and  to  range 

OUTLINE  OF  FINISHED  SHEET 


V- 

N                                                                                                                 > 

1 

BORDER  LINE  OF  SHEET 

1  ' 

V 

1  1 

'?' 

•t  nr> 

o 

*           NOTATION  TO  BE  USED 

POINTS            LINES                                      PLA 

NES 

Visible     Invisible                   Traces 

Intersections 

REQUIRED 

®        ' 



f^ 

PROJECTING 



j 

^ 

. 

; 

> 

£ 

2 

* 

FIG.  76. 

in  size  from  J"  to  -fa"  high.  Any  lettering  may  be  condensed, 
square,  or  extended. 

The  sheets  must  be  well  balanced,  the  figures  with  reference 
to  one  another,  and  the  whole  with  reference  to  the  border-line. 
(Do  not  fail  to  reserve  a  space  at  the  top  for  the  title  of  the  sheet, 
and  a  space  at  the  lower  right-hand  corner  for  the  signature.) 
(See  sample  sheets,  Plates  i  and  2,  pages  99  and  100). 

Each  sheet  is  to  be  neatly  executed  in  pencil,  in  accordance 
with  the  foregoing  directions,  then  submitted  for  approval,  and 
when  approved,  inked  in,  and  lettered,  cleaned  and  trimmed  to 
size  (9"Xi2")>  then  offered  for  acceptance,  grading,  and  filing. 


THEORETICAL   PROBLEMS. 


99 


PLATE  No.  i. 


CO 

o 

o: 

CL 

o 


cr 
o 


IOO 


ADVANCED  MECHANICAL   DRAWING. 


PLATE  No.  2. 


09 

d 

z 

^ 

a 

CD 
O 
DC 
DL 

.J 
O 


o: 
o 


ed. 


THEORETICAL   PROBLEMS.  101 


POINTS,   LINES,   AND   PLANES. 

53.  PROBLEM  i: 

(a)  Locate  the  point  P  in  the  first  quadrant  f"  from  V 
and  i"  jrom  H. 

(b)  Locate  the  point  Q  in  the  second  quadrant  \"  jrom  V 
and  I"  jrom  H. 

(c)  Locate  the  point  R  in  the  third  quadrant  J"  jrom  V 
and  £"  jrom  H. 

(d)  Locate  the  point  S  in  the  fourth  quadrant  i"  jrom  V 
and  \"  jrom  H. 

(e)  Show  a  line  in  the  first  quadrant,  passing  through  the 
fourth  quadrant  into  the  third  quadrant. 

(f)  Show  a  line  in  the  first  quadrant  2"  long,  30°  to  F, 
and  parallel  to  H. 

Suggestion : 

(a),  (6),  (c),  (d).  Since  the  location  of  a  point  with  reference 
to  H  is  shown  by  its  V  projection,  and  its  position  with  reference 
to  V  is  shown  by  its  H  projection,  and  since  the  two  projections 
of  a  point  always  lie  in  the  same  perpendicular  to  the  ground- 
line,  draw  a  line  perpendicular  to  the  ground-line,  and  on  it  lay 
off  the  dimensions  of  the  point. 

(e)  Assume  one  end  of  the  line  in  the  first  quadrant  (a 
point),  and  a  point  in  the  fourth  quadrant;  join  the  like  pro- 
jections of  these  two  points,  and  produce  the  line  obtained 
into  the  third  quadrant :  this  line  will  be  the  required  line. 

(/)  Since  the  position  of  a  line  with  reference  to  V  is  shown 
by  its  H  projection,  and  since  a  line  is  projected  in  its  true  length 
on  a  parallel  plane,  draw  a  line  in  the  H  plane  2"  long  and  mak- 
ing an  angle  of  30°  with  the  ground-line:  this  line  will  be  the  H 
projection  of  the  line;  to  locate  its  vertical  projection,  draw  per- 
pendiculars to  the  ground-line  through  the  extremes  of  the  H 
projection,  and  at  any  convenient  point  above  the  ground-line 
draw  a  horizontal  line  between  the  perpendiculars:  this  line  will 
be  the  required  V  projection. 


102  ADVANCED  MECHANICAL  DRAWING. 

54.  PROBLEM  2: 

(a)  Show  a  line  in  the  first  quadrant,  2"  long,  the  projections 
of  which  are  30°  to  the  ground-line. 

(b)  Find  the  projections  oj  a  line  (in  any  quadrant),  2" 
long,  45°  to  F,  and  30°  to  H. 

(c)  Show  a  plane,  T,  oblique  to  the  ground-line,  and  draw 
a  line  in. the  plane. 

(d)  Find  the  traces  oj  a  plane,  S,  that  will  contain  the  line 
drawn  in  plane  T  (c). 

Suggestion : 

(a)  Assume  the  projections  of  any  line,  the  projections  of 
which  are  at  30°  to  the  ground-line,  then  revolve  the  line  into 
coincidence  with  one  of  the  planes  of  projection;   in  this  position 
the  line  will  appear  in  its  true  length,  therefore  lay  off  a  2"  length 
on  the  revolved  line,  then  revolve  the  line  back  to  its  original 
position. 

(b)  Assume  a  line  2"  long  and  parallel  with  the  ground- 
line  (in  this  position  the  projections  will  each  be  2"  long  and 
parallel  to  the  ground-line) ;    now,  revolve  the  H  projection  of 
this  line  about  one  extreme  until  it  makes  the  V  angle  with  the 
ground-line;    the  new  position  of  the  other  extreme  defines  the 
locus  of  the  point  in  the  required  projection  with  reference  to 
the  vertical  plane.     Next,  revolve  the  V  projection  of  the  parallel 
line  about  the  same  extremes  as  used  above,  until  it  makes  the 
H  angle  with  the  ground-line;    this  position  defines  the  locus 
of  the  other  extremes  of  the  line  in  the  required  projection  with 
reference  to  the  horizontal  plane;    revolve  the  H  projection  of 
this  extreme  in  this  position  until  it  intersects  a  horizontal  line 
drawn  through  the  point  defining  the  locus  of  the  point  with  refer- 
ence to  F;    this  intersection  will  be  the  required  H  projection 
of  one  extreme  of  the  line;   the  distance  of  the  required  vertical 
projection  of  the  point  being  defined  by  the  point  showing  the 
distance  of  the  required  projection  from  H,  the  V  projection 
is  obtained  by  projection;   this  extreme  projected,  join  the  pro- 
jections with  the  projections  of  the  other  extreme  (this  point  has 


THEORETICAL   PROBLEMS. 


remained  stationary),  and  the  resulting  lines  will  be  the  required 
projections. 

(c)  Since  a  plane  is  shown  by  its  intersection  with  H  and 
V  (its  traces),  a  plane  that  is  oblique  to  the  ground-line  will 
have  its  traces  oblique  to  the  ground-line.    To  locate  a  line  in 
any  given  plane  it  is  only  necessary  to  assume  a  point  in  each 
trace  and  to  then  join  these  two  points  with  a  line. 

(d)  To  pass  a  plane  through  any  given  line,  find  where 
the  line  pierces  the  planes  of  projection.  These  points  will  be  points 
in  the  respective  traces,  and  since  the  traces  of  a  plane  always 
meet  in  a  point  in  the  ground-line,  assume  any  point  in  the  ground- 
line  and  join  it  with  the  points  in  which  the  line  pierces  V  and  H. 

55.  PROBLEM  3: 

(a)  Revolve  the  point  O  about  the  line  M-N  into  V.     (Fig. 

770 

(b)  Pass  a  plane  through  three  points,  M-N-O.     (Fig.  78.) 

(c)  Find  the  intersection  o]  two  planes,  T  and  S.     (Fig.  79.) 

•n 


On' 


•m 


m 


t° 


•O' 


FIG.  78. 


Suggestion : 

(a)  Pass  a  plane  through  the  point  perpendicular  to  the 
line;  find  the  intersection  of  this  plane  with  V,  and  with  the  point 
at  which  the  line  pierces  this  plane  as  a  center,  revolve  the  point 
into  the  V  trace  of  the  auxiliary  plane. 

(6)  Draw  a  line  through  any  two  of  the  points,  and  through 
any  point  in  this  line  and  the  remaining  point  draw  a  second  line; 


104  ADVANCED  MECHANICAL  DRAWING. 

find  where  these  lines  pierce  H  and  F,  and  draw  the  V  trace 
of  the  required  plane  through  the  two  points  in  which  the  lines 
pierce  F,  and  the  H  trace  through  the  two  points  in  which  the 
lines  pierce  H. 

(c)  Since  the  V  traces  intersect  in  a  point,  and  the  H 
traces  intersect  in  another  point,  there  will  be  two  points  in  the 
intersection,  which  is  a  right  line.  Draw  the  line  of  intersection 
through  these  two  points. 


r 


m 


FIG.  79.  FIG.  80. 


56.  PROBLEM  4: 

(a)  Pass  a  plane  through  the  point  P  perpendicular  to  the 
line  M-N.     (Fig.  80.) 

(b)  Find  the  perpendicular  distance  from  the  point  P  to 
the  line  M-N.     (Fig.  81.) 

Suggestion : 

(a)  The  traces  of  the  required  plane  are  perpendicular 
to  the  corresponding  projections  of  the  given  line;  therefore, 
through  the  point  draw  a  line  parallel  to  either  trace,  and  find 
the  point  in  which  it  pierces  -H"  or  F;  this  is  one  point  in  the  trace 
on  this  plane;  through  this  point  draw  a  line  perpendicular  to 
the  corresponding  projection  of  the  given  line,  and  from  the 
point  in  which  it  crosses  the  ground-line  draw  a  line  perpen- 
dicular to  the  other  projection  of  the  lines.  These  lines  will  be 
the  traces  of  the  required  plane. 


THEORETICAL   PROBLEMS. 


(b)  First  method. — Pass  a  plane  through  the  point  and 
the  given  line,  and  revolve  the  plane  into  one  of  the  planes  of 
projection  about  the  corresponding  trace,  then  draw  a  line  from 
the  revolved  position  of  the  point  perpendicular  to  the  revolved 
position  of  the  line;  this  is  the  required  distance. 

Second  method. — Pass  a  plane  through  the  point  per- 
pendicular to  the  line;  find  where  the  line  pierces  this  plane 
and  join  this  point  with  the  given  point.  The  true  length  of  this 
line  is  the  required  perpendicular  distance. 


57- 


PROBLEM  5: 

(a)  Find  where  a  line  pierces  a  given  plane. 

(b)  Project  the  line  M-N  on  the  plane  T.     (Fig.  82.) 

(c)  Project  the  ground-line  on  the  plane  T.     (Fig.  82.) 


m' 


m 


FIG.  82. 


Suggestion : 

(a)  Pass  a  plane  through  the  given  line  perpendicular  to 
one  of  the  planes  of  projection,  and  find  the  intersection  of  the 
plane  with  the  given  plane.    The  point  in  which  the  line  pierces 
the  plane  must  evidently  be  a  point  in  the  intersection  of  these 
two  planes,  also,  the  point  must  lie  in  the  projection  of  the  line; 
therefore,  the  intersection  of  these  two  lines  is  the  required  point. 

(b)  Drop  a   perpendicular  from  each  end  of  the   given 
line  to  the  given  plane,  and  find  where  these  lines  pierce  the  plane; 
join  the  points  by  a  right  line. 


io6 


ADVANCED  MECHANICAL  DRAWING. 


(c)  Where  the  traces  of  the  given  plane  meet  in  the  ground- 
line  will  be  one  point  in  the  required  projection;  from  any  other 
point  in  the  ground-line  drop  a  perpendicular  to  the  given  plane 
and  find  where  it  pierces  it;  join  this  point  with  the  intersection 
of  the  traces  on  the  ground-line. 

58.  PROBLEM  6: 

(a)  Find  the  angle  between  the  line  M-N  and  the  plane  T. 
(Fig.  83.) 

(b)  Through  the  point  P  draw  a  line  making  an  angle  of 
45°  with  the  plane  T.     (Fig.  83.) 

Suggestion : 

(a)  First  method. — The  angle  between  a  line  and  a  plane 
is  the  angle  between  the  line  and  its  projection  on  the  plane; 
therefore,  project  the  given  line  on  the  given  plane,  pass  a  plane 


FIG.  84. 

through  the  line  and  its  projection,  and  revolve  this  plane  into 
one  of  the  planes  of  projection.  The  angle  between  the  lines 
in  the  revolved  position  is  the  required  angle. 

Second  method. — Drop  a  perpendicular  from  any  point 
in  the  given  line  to  the  given  plane;  the  angle  between  the  per- 
pendicular and  the  given  line  will  be  the  complement  of  the 
required  angle. 

(6)  Pass  a  plane  through  the  given  point  perpendicular 


THEORETICAL  PROBLEMS.  107 

to  the  given  plane,  and  find  its  intersection  with  the  given  plane; 
revolve  the  perpendicular  plane  about  one  of  its  traces,  and  draw 
a  line  through  the  revolved  position  of  the  point  making  the 
required  angle  with  the  revolved  position  of  the  intersection  of 
the  two  planes;  now  revolve  this  line  back  to  the  original  posi- 
tion of  the  point,  this  will  give  the  projections  of  the  required 
line. 

59.  PROBLEM  7: 

(a)  Find  the  angle  between  the  planes  T  and  S.     (Fig.  84.) 

(b)  Find  the  traces  oj  a  plane  T  making  an  angle  oj  60° 
with  H  and  45°  with  V. 

Suggestion : 

(a)  Through  any  convenient  point  in  the  line  of  inter- 
section of  the  given  planes  pass  a  plane  perpendicular  to  the  line 
of  intersection;    this  plane  will  cut  a  line  from  each  plane;    the 
angle  between  these  two  lines  is  the  required  angle.     To  show 
the  true  size  of  this  angle,  revolve  the  auxiliary  plane  about  one 
of  its  traces  into  the  corresponding  plane  of  projection. 

(b)  Consider  the  problem  solved :  now,  through  any  point, 
P,  in  the  ground-line  pass  a  plane,  R,  perpendicular  to  the  H 
trace  of  T — this  plane  will  give  the  H  angle;   next  pass  a  plane 
through  P  perpendicular  to  the  V  trace  of  T — this  gives  the 
V  angle.     The  perpendicular  distance  from  P  to  the  intersection 
of  either  planes  R  and  T  or  5  and  T  is  the  perpendicular  dis- 
tance from  P  to  plane  T\   therefore,  to  find  the  traces  of  a  plane 
making  given  angles  with  H   and  V   assume  any  point  in  the 
ground-line  as  point  P,  and  with  this  as  a  center  and  any  con- 
venient radius,  as  the  perpendicular  distance  from  P  to  T,  describe 
a  circle;    tangent  to  this  and  above  the  ground-line  draw  a  line 
making  the  given  H  angle  with  the  ground-line;  the  point  where 
this  line  cuts  a  perpendicular  to  the  ground-line  through  point  P 
will  be  one  point  in  the  V  trace,  and  the  distance  from  P  to  where 
this  line  cuts  the  ground-line  will  be  the  radius  of  a  circle  to  which 
the  H  trace  will  be  tangent;  now,  tangent  to  the  first  circle  and 
below  the  ground-line  draw  a  line  making  the  V  angle  with 


io8 


ADVANCED  MECHANICAL  DRAWING. 


the  ground-line;  where  this  line  cuts  the  perpendicular  through 
P  will  be  one  point  in  the  H  trace;  through  this  point  and  tan- 
gent to  the  second  circle  draw  the  H  trace,  and  from  where  it 
cuts  the  ground-line  draw  the  V  trace  through  the  V  point  in 
the  trace. 
60.  PROBLEM  8: 

(a)  Find    the    perpendicular   distance    between    two    lines, 
M-N  and  O-P.     (Fig.  85.) 

(b)  Show  two  parallel  planes  that  are  \"  apart. 


o 

FIG.  85. 
Suggestion : 

(a)  First  method. — Through  one  line  pass  a  plane  parallel 
to  the  second  line,  and  project  the  second  line  on  the  plane; 
at  the  point  where  this  projection  intersects  the  first  line  erect  a 
perpendicular  to  the  plane,  and  produce  it  until  it  intersects 
the  second  line.  The  true  length  of  this  perpendicular  is  the 
required  distance. 

Second  method. — Revolve  the  two  lines  about  a  point 
in  one  of  the  lines  (usually  one  extreme)  until  that  line  is  parallel 
to  F;  next,  revolve  the  lines  in  this  position  about  the  same 
point  until  the  line  parallel  to  F  is  perpendicular  to  H.  This  line 
will  now  project  on  H  as  a  point,  and  the  second  line  as  a  line. 
Now  draw  a  line  through  the  H  projection  of  the  point  perpen- 
dicular to  the  H  projection  of  the  line;  this  will  be  the  required 
perpendicular  distance  between  the  two  lines. 


THEORETICAL  PROBLEMS. 


109 


(b)  Assume  one  of  the  planes  as  a  given  plane,  and  pass 
an  auxiliary  plane  perpendicular  to  it.  This  plane  will  cut  a  line 
from  the  given  plane  and  a  parallel  one  from  the  required  plane, 
and  the  distance  between  the  lines  will  be  the  perpendicular 
distance  between  the  planes;  therefore,  revolve  the  auxiliary 
plane  about  one  of  its  traces  into  the  corresponding  plane  of 
projection,  and  draw  the  line  of  the  required  plane  the  given 
distance  from  the  line  cut  from  the  given  plane.  The  points  in 
which  this  line  pierces  H  and  V  are  points  in  the  corresponding 
trace  of  the  required  plane;  through  these  points  draw  traces 
parallel  to  the  respective  traces  of  the  given  plane. 

61.  PROBLEM  9: 

Pass  a  circle  through  three  points.     (Fig.  86.) 


FIG.  86. 


FIG.  87. 


Suggestion : 

Pass  a  plane  through  the  three  points  and  revolve  the 
plane  into  H  or  V  about  the  corresponding  trace;  the  points 
will  then  appear  in  their  true  position  with  respect  to  one  another; 
therefore,  draw  a  circle  through  the  three  points  while  in  this 
position,  assume  a  number  of  points,  other  than  the  three  given 
points  on  the  circle,  and  revolve  the  plane  back  to  its  initial 
position. 


no  ADVANCED  MECHANICAL  DRAWING. 

TANGENT  PLANES. 

62.  PROBLEM  10: 

(a)  Pass  a  plane  tangent  to  a  cone  at  a  point  on  the  surface. 
(Fig.  87.) 

(b)  Pass  a  plane  parallel  to  a  line  M-N  and  tangent  to  a 
cylinder.     (Fig.  88.) 


FIG.  88. 


FIG.  89. 


Suggestion : 

(a)  Through  the  point  on  the  cone  draw  an  element  of 
the  cone;    at  the  point  where  this  element  cuts  the  base  of  the 
cone  draw  a  tangent  to  the  base.    These  two  intersecting  lines 
will  determine  the  tangent  plane. 

(b)  Through  any  point  in  the  line  draw  a  line  parallel  to 
the  elements  of  the  cylinder;    these  two  lines  will  determine  a 
plane  parallel  to  the  required  plane;    this  plane  will  cut  a  line 
from  the  plane  of  the  base  of  the  cylinder;   parallel  to  this  line 
draw  a  line  tangent  to  the  base  of  the  cylinder,  and  at  the  point 
of  tangency  draw  an  element  of  the  cylinder.   These  two  lines  will 
determine  the  required  plane. 

63.  PROBLEM  n: 

(a)  Pass  a  plane  tangent  to  a  cylinder  at  a  point  on  the 
surface.     (Fig.  89.) 

(b)  Pass  a  plane  through  a  point  without  the  cone  tangent 
to  the  cone.     (Fig.  90.) 


THEORETICAL  PROBLEMS. 


in 


Suggestion : 

(a)  Through  the  point  on  the  cylinder  draw  an  element; 
at  the  point  where  the  element  cuts  the  base  of  the  cylinder  draw 
a  tangent  to  the  base.    These  two  lines  will  determine  the  tangent 
plane. 

(b)  Draw  a  line  through  the   given  point  and  the   apex 
of  the  cone;   find  where  the  line  pierces  the  plane  of  the  base  of 
the  cone,  and  through  this  point  draw  a  line  tangent  to  the  base. 
These  two  lines  will  determine  the  required  plane. 

64.  PROBLEM  12: 

(a)  Pass  a  plane  tangent  to  a  sphere  at  a  point  on  the  sur- 
face. 

(b)  Pass  a  plane  parallel  to  a  line  and  tangent  to  a  cone. 
(Fig.  91.) 


•P 


FIG.  90. 


FIG.  91. 


Suggestion : 

(a)  Pass  a  plane  through  the  point  parallel  to  either  H 
or  F,  and  draw  a  tangent  to  the  curve  cut  from  the  surface  at 
the  given  point.  This  line  will  pierce  one  of  the  planes  of  projec- 
tion, and  through  this  point  draw  a  trace  perpendicular  to  the 
corresponding  normal  to  the  surface  at  the  point  of  tangency; 
from  where  this  trace  cuts  the  ground-line  draw  the  other  trace 
perpendicular  to  the  other  projection  of  the  normal. 


112 


ADVANCED  MECHANICAL  DRAWING. 


(b)  Draw  a  line  through  the  apex  of  the  cone  parallel  to 
the  given  line;  produce  this  line  to  an  intersection  with  the  plane 
of  the  base  of  the  cone,  and  through  the  point  of  intersection 
draw  a  line  tangent  to  the  base.  These  two  lines  will  determine 
the  required  plane. 

65.  PROBLEM  13: 

(a)  Pass  a  plane  tangent  to  a  hyperbolic  paraboloid  at  a 
point  on  the  surface.     (Fig.  92.) 

(b)  Pass  a  plane  tangent  to  a  cylinder  and  through  a  point 
P  outside  of  the  cylinder.     (Fig.  93.) 


•P' 


FIG.  92. 


FIG.  93. 


Suggestion : 

(a)  Pass  a  projecting  plane  through  the  point.    This  plane 
will  cut  a  curve  from  the  surface;  a  line  drawn  tangent  to  this 
curve  at  the  given  point  will  be  one  line  in  the  required  plane. 
Pass  some  other  projecting  plane  and  get  a  second  curve  and 
tangent  line  at  the  point  on  the  surface;    the  two  tangent  lines 
will  determine  the  required  plane. 

(b)  Draw  a  line  through  the  point  and  parallel  to  the  axis 
of  the  cylinder  and  find  where  it  pierces  the  plane  of  the  base 
of  the  cylinder;    through  this  point  draw  a  line  tangent  to  the 
base.     This  line  and  the  line  through  the  point  will  determine 
the  required  plane. 


THEORETICAL  PROBLEMS. 


66.  PROBLEM  14: 

(a)  Pass  a  plane  through  a  line  and  tangent  to  a  sphere. 
(Fig.  94.) 

(b)  Pass  a  plane  perpendicular  to  a  line  and  tangent  to  a 
sphere.     (Fig.  95.) 


FIG.  94.  FIG.  95. 

Suggestion : 

(a)  Single-cone  method. — Take  a  line  through  the  center 
of  the  sphere  and  parallel  to  either  H  or  V,  and  produce  this  line 
to  an  intersection  with  the  given  line.    This  line  will  be  the  axis 
of  a  tangent  cone,  and  the  point  where  it  intersects  the  given 
line  will  be  the  apex  of  the  cone.     The  base  of  the  cone  will  be 
perpendicular  to  the  axis.     Find  where  the  given  line  pierces  the 
plane  of  the  base,  and  from  this  point  draw  a  line  tangent  to  the 
base.     This  line  and  the  given  line  will  determine  the  required 
plane. 

(b)  Draw  a  line  through  the  center  of  the  sphere  parallel 
to  the  given  line,  and  find  where  it  pierces  the  surface  of  the 
sphere;    pass  a  plane  through  this  point  perpendicular  to  the 
given  line. 

67.  PROBLEM  15: 

(a)  Pass  a  plane  tangent  to  a  helical  convolute  and  parallel 
to  a  line.     (Fig.  96.) 

(b)  Pass  a  plane  tangent  to  a  helicoid  and  perpendicular 
to  a  line.     (Fig.  97.) 


H4  ADVANCED  MECHANICAL  DRAWING. 

Suggestion : 

(a)  With  any  point  in  the  line  as  the  apex  draw  a  cone 
the  elements  of  which  make  the  same  angle  with  the  H  plane  as 


m' 


FIG.  96. 


FIG.  97. 

do  the  elements  of  the  convolute.  Pass  a  plane  through  the  line 
tangent  to  this  auxiliary  cone,  and  find  the  element  of  tangency; 
now  find  an  element  of  the  convolute  parallel  to  this  element 


THEORETICAL  PROBLEMS. 


of  the  cone,  and  find  where  it  pierces  H  and  F;  draw  the  re- 
quired traces  through  these  points  parallel  to  the  respective 
traces  of  the  plane  tangent  to  the  cone. 

(b)  With  any  point  in  the  vertical  projection  of  the  axis 
of  the  helicoid  as  the  apex  construct  a  cone  the  elements  of  which 
make  the  same  angle  with  H  as  do  the  elements  of  the  helicoid, 
and  pass  a  plane  through  the  apex  of  this  cone  parallel  to  the 
required  plane,  that  is,  perpendicular  to  the  given  line.  This 
plane  will,  in  most  cases,  cut  two  elements  from  the  cone;  now 
find  an  element  in  the  helicoid  parallel  to  either  element  cut  from 
the  cone,  and  find  the  points  in  which  it  pierces  H  and  F;  through 
these  points  draw  traces  parallel  to  the  respective  traces  of  the 
auxiliary  plane.  This  plane  will  be  the  required  plane. 

68.  PROBLEM  16: 

Pass  a  plane  through  a  line  and  tangent  to  any  double  curved 
surface  oj  revolution.  (Fig.  98.) 


FIG.  98.  FIG.  99. 

Suggestion : 

Revolve  the  line  around  the  axis  of  the  surface  and  generate 
a  rolling  hyperboloid  an  auxiliary  surface  of  revolution;  now 
pass  a  meridian  plane  through  the  axis  parallel  to  the  plane  of 
projection  to  which  the  axis  is  parallel;  this  plane  will  cut  a 
meridian  curve  from  each  surface;  next  draw  a  common  tan- 
gent to  these  two  curves,  then  revolve  it  about  the  axis  until 


u6 


ADVANCED  MECHANICAL  DRAWING. 


it  intersects  the  given  line.     These  two  intersecting  lines  will  deter- 
mine the  required  plane. 

INTERSECTIONS. 

(a)  Find  the  intersection  oj  a  cone  and  a  plane.     (Fig.  99.) 

69.  PROBLEM  17: 

(b)  Find  the  intersection  0}  a  cone  and  a  cylinder.     (Fig.  100.) 


FIG.  zoo. 


FIG.  101. 


Suggestion: 

(a)  Pass  a  series  of  projecting  planes  through  the  apex 
of  the  cone  that  will  cut  elements  from  the  cone  and  right  lines 
from  the  plane.      The  intersection  of  these  lines  will  be  points 
in  the  required  curve  of  intersection. 

(b)  Draw  a  line  through  the  apex  of  the  cone  parallel  to 
the  axis  of  the  cylinder,  and  pass  planes  through  it  that  will 
cut  elements  from  both  the  cone  and  cylinder.     The  intersection 
of  these  lines  will  be  points  in  the  required  intersection. 

70.  PROBLEM  18: 

(a)  Find  the  intersection  of  a  cylinder  and  a  plane.     (Fig. 
101.) 

(b)  Find  the  intersection  of  two  cones.     (Fig.  102.) 

Suggestion : 

(a)  Pass  a  series  of  projecting  planes  through  the  elements 


THEORETICAL   PROBLEMS.  117 

of  the  cylinder;  these  planes  will  cut  right  lines  from  the  given 
plane.  The  intersection  of  these  lines  with  the  elements  of  the 
cylinder  will  be  points  in  the  required  intersection. 


7 


FIG.  102.  FIG.  103. 

(b)  Draw  a  line  through  the  apices  of  the  two  cones,  and 
pass  a  series  of  planes  through  it  that  will  cut  elements  from 
both  cones.  The  intersections  of  the  elements  will  be  points  in 
the  required  intersection. 

71.  PROBLEM  19: 

(a)  Find  the  intersection  oj  a  sphere  with  a  plane.     (Fig. 
103.) 

(b)  Find  the  intersection  oj  two  cylinders.     (Fig.  104.) 

Suggestion : 

(a)  Pass  a  series  of  projecting  planes  through  the  sphere; 
these  planes  will  cut  circles  from  the  sphere  and  right  lines  from 
the  given  planes.    The  intersection  of  these  lines  with  the  circles 
will  be  points  in  the  required  intersection. 

(b)  At  some  convenient  point  assume  a  point,  and  through 
it  draw  a  line  parallel  to  the  elements  of  each  cylinder;    these 
two  lines  will  determine  a  plane  director;    now  pass  a  series  of 
planes   parallel  to  this  auxiliary   plane   that  will  cut   elements 
from  both  cylinders.    The  intersections  of  these  elements  will  be 
points  in  the  required  intersection. 


n8 


ADVANCED  MECHANICAL  DRAWING. 


DEVELOPMENTS. 
72,  PROBLEM  20: 

Develop  the  oblique  coney  Fig.  105. 
Suggestion : 

Draw  a  number  of  elements  of  the  cone,  and  find  the  true 
length  of  each.  To  begin,  select  some  particular  element,  usually 
the  longest  or  the  shortest,  and  lay  this  off  in  its  true  length; 
next,  take  the  distance  from  the  point  where  this  element  cuts 
the  base  of  the  cone  to  the  point  where  the  next  element  cuts  the 
base,  as  a  radius,  and  with  one  end  of  the  line  already  laid  off 


FIG.  104. 


FIG.  105. 


as  a  center,  describe  an  arc;  now,  with  the  other  end  of  the  de- 
veloped element  as  the  developed  apex  of  the  cone,  as  a  center, 
and  a  radius  equal  to  the  true  length  of  the  second  element, 
strike  a  second  arc  intersecting  the  first  one,  the  line  joining  this 
point  and  the  apex  will  be  the  developed  position  of  the  second 
element;  proceed  in  this  manner,  taking  one  element  at  a  time, 
until  all  of  them  have  been  laid  out,  then  draw  a  curved  line 
through  the  free  end  of  the  elements.  The  figure  obtained  will 
be  the  developed  cone. 

73.  PROBLEM  21: 

Develop  the  oblique  cylinder,  Fig.  106. 
Suggestion : 

Revolve  the  cylinder  until  it  is  parallel  to  either  H  or  F, 
and  while  in  this  position  pass   a   plane  perpendicular  to  the 


THEORETICAL  PROBLEMS. 


119 


cylinder;  this  plane  will  cut  a  right  section  from  the  cylinder 
that  will  develop  as  a  right  line;  next,  assume  a  number  of  ele- 
ments, and  find  the  true  size  of  the  right  section;  this  will  show 
the  true  distance  between  the  elements;  now  lay  off  the  right 
section  in  its  true  development,  and  locate  the  points  where  the 
elements  cut  it;  draw  lines  through  these  points  perpendicular 
to  the  right  line  of  the  developed  section,  and  on  these  lay  off 
the  true  lengths  of  the  elements  on  each  side  of  the  section;  finish 
the  development  by  drawing  curved  lines  through  the  ends  of 
the  elements. 

74.  PROBLEM  22: 

Develop  the  cylinder  between  the  horizontal  plane  and  tfte 
hyperbolic  paraboloid,  Fig.  107. 


FIG.  106. 


FIG.  107. 


Suggestion : 

In  this  case  the  cylinder  is  perpendicular  to  H.  First 
find  the  intersection  of  the  cylinder  with  the  surface;  this  will 
give  the  upper  base  of  the  cylinder;  now  the  horizontal  base  of 
the  cylinder  is  a  right  section,  and  will  develop  as  a  right  line, 
and  the  vertical  projection  of  the  cylinder  shows  the  true  lengths 
of  the  elements;  therefore,  proceed  as  in  Problem  21. 


I2O 


ADVANCED  MECHANICAL  DRAWING. 


75.  PROBLEM  23: 

Develop  the  general  case  of  the  convolute  surface.     (Fig.  108.) 

Suggestion : 

Since  the  convolute  surface  is  a  single  curved  surface  a 
plane  can  be  passed  through  any  two  consecutive  elements; 
therefore,  assume  a  number  of  elements  of  the  surface  as  close 

together  as  is  practicable,  and  thus 
divide  the  surface  into  a  number  of 
small  sections;  now,  beginning  with 
any  section,  revolve  it  about  a  trace 
drawn  through  the  points  in  which 
the  two  limiting  elements  pierce  the 
horizontal  plane  until  each  element 
is  coincident  with  H ;  this  will  give 
the  approximate  size  of  the  section; 
next,  repeat  this  process  for  all  of 
FlG-  Io8-  the  small  sections,  then  add  these 

developments  together,  and  draw  a  curved  line  through  each 
end  of  the  elements.  The  figure  thus  obtained  is  the  required 
development,  approximately. 

76.  PROBLEM  24. 

Develop  a  2"  sphere,  approximately. 

Suggestion : 

Pass  a  series  of  meridian  planes  cutting  the  sphere  into  a 
number  of  equal  meridian  sections,  and  develop  one  section 
carefully ;  repeat  for  the  number  of  sections  in  the  surface. 


CHAPTER  V. 
PRACTICAL  PROBLEMS. 

77.  Explanatory. — The  student   having  acquired   a  working 
knowledge  of  the  principles  of  Descriptive  Geometry,  the  follow- 
ing examples  are  given  to  illustrate  their  practical  application. 
The  examples  offered  are  typical  of  the  problems  confronting 
the   engineering   draughtsman   in   every-day   practice,   with   the 
exception,  however,  that  they  are  more  or  less  hypothetical  in 
that  all  Design  is  omitted  (no  allowance  is  made  for  lap,  fasten- 
ings, etc.),  the  work  being  preliminary  to  that  subject. 

78.  General    Directions. — The    general    directions    for    the 
execution  of  the  exercises  are  the  same  as  those  given  in  Chapter 
IV,  page  97,  the  specific  differences  being  that  here  each  exercise 
has  all  necessary  dimensions  given,  either  on  the  plate  or  in  the 
instructions.   Some  of  the  exercises  are  to  be  drawn  full  size,  others 
to  some  proportional  scale.     For  some  exercises  dimensions  for 
balancing  the  drawing  on  the  sheet  are  given.    In  every  case  these 
figures  represent  full-size  lengths,  and  are  to  be  omitted  on  the 
finished  drawing. 

In  place  of  the  notation  given  in  Cha'pter  IV,  use  the  con- 
ventions of  ordinary  drawing,  and  make  all  working  lines  very 
light,  full  lines. 

121 


122  ADVANCED  MECHANICAL  DRAWING. 

TRUE  LENGTHS,   TRUE  ANGLES,   INTERSECTIONS, 
DEVELOPMENTS,  ETC. 

79.  PROBLEM  i: 

To  lay  out  the  cutting  lines  jor  gelling  out  the  wreath  start- 
ing from  a  newel  post. 

Let  the  problem  be  that  presented  by  Plate  3,  and  let  it 
be  required  to  show  the  layout  in  isometric.  As  may  be  seen 
by  the  plate,  the  wreath  is  the  curved  portion  of  the  hand-rail. 
Such  a  piece  is  usually  cut  from  a  block;  to  find  the  size  of  the 
block,  inclose  the  mechanical  drawings — plan  and  elevation — 
of  the  wreath  within  rectangles;  the  length  of  the  block  will 
equal  the  length  of  either  rectangle — the  lengths  are  the  same — 
the  width  will  equal  the  width  of  the  rectangle  inclosing  the  plan 
drawing,  and  the  thickness  of  the  block  will  equal  the  width  of 
the  rectangle  inclosing  the  elevation  drawing.  In  laying  out 
the  cutting  lines  on  the  block,  the  top  face  will  contain  the  plan 
of  the  wreath,  and  the  right  side  the  elevation  drawing;  these 
lines  laid  out,  the  wreath  may  be  sawed  out  by  cutting  along 
them.  The  drawing  shows  a  rectangular  rail;  in  practice  the 
rail  is  of  a  section  calculated  to  be  ornamental,  and  useful  as  a 
grip  for  the  hand;  such  a  section  is  carved  in  the  wreath  after 
cutting  out  as  above. 

The  isometric  drawing  is  constructed  according  to  the 
principles  of  isometric  drawing  as  set  forth  in  Chapter  I,  by  first 
drawing  the  outline  of  the  inclosing  block,  then  locating  the 
wreath  within  it. 

Directions  for  Drawing. 

Execute  a  scale  drawing  (i"  =  i")  according  to  the  dimen- 
sions given,  drawing  the  plan  first,  then  the  elevation,  then  lay 
out  the  block.  Draw  all  necessary  lines  in  light  pencil,  then 
submit  the  drawing  for  inspection.  In  inking,  ink  only  those 
lines  shown  on  the  plate,  give  all  dimensions — supplying  those 
marked  X — and  finish  the  sheet  by  lettering  it  as  shown. 


PRACTICAL  PROBLEMS. 


1*3 


PLATF.  No    .3 


124  ADVANCED  MECHANICAL  DRAWING. 

80.  PROBLEM  2: 

To  show  the  layout  jor  the  shop,  for  a  wrought-iron  corner 
support. 

Let  the  problem  be  that  presented  by  the  mechanical 
drawing  of  Plate  4,  illustrating  a  plan  and  elevation  of  a  wrought- 
iron  corner  support  for  an  all-steel  fence.  An  inspection  of  the 
drawing  shows  the  support  to  be  in  one  piece,  and  of  a  shape 
including  both  plane  and  single  curved  surfaces,  and  therefore 
developable. 

To  develop  the  piece,  draw  the  center  line  A-E  of  the  plan 
drawing,  then  draw  the  center  line  C-D  of  the  development  and 
on  it  lay  off  horizontal  lengths  corresponding  to  the  vertical 
dimensions  of  the  elevation  drawing — these  lengths  being  there 
shown  in  their  true  dimensions  since  the  lines  are  parallel  to  that 
plane  of  projection;  next,  draw  the  indefinite  base  line  E-F 
perpendicular  to  C-D,  and  working  from  the  center-point  G 
of  the  plan,  take  dimensions  by  stepping  along  the  center  line 
A-E — both  ways — and  lay  them  off  on  E-F  symmetrical  with 
the  center  line  C-D',  through  this  last  set  of  points,  draw  a  series 
of  horizontal  lines,  and  through  the  first  set  of  points — those  on 
the  line  C-D — draw  a  series  of  vertical  lines;  these  two  series 
of  lines  will  intersect  in  a  series  of  points  from  which  the  outline 
of  the  development  may  be  obtained. 

In  addition  to  the  above  statement,  let  it  be  required  to 
elucidate  the  mechanical  drawing  with  an  isometric  drawing  of 
the  support.  This  is  done  by  first  assuming  the  mechanical 
drawing  to  be  inclosed  within  a  rectangular  box,  then  draw  the 
box  as  suggested  by  the  light  lines  of  the  isometric  drawing,  and 
in  accordance  with  the  principles  of  isometric  drawing  as  set 
forth  in  Chapter  I,  construct  the  piece  within  the  box,  then  erase 
the  working  lines. 

Directions  for  Drawing. 

Execute  a  full  size  drawing  according  to  the  dimensions 
given,  drawing  the  plan  first,  then  the  elevation,  then  lay  out 
the  development;  these  drawn,  execute  the  isometric  drawing. 


PRACTICAL   PROBLEMS, 


I25 


PLATE  No.  4. 


I  (-xoq  3?uis\p-ui  jo  aui|  ;UQJJ) 


126  ADVANCED  MECHANICAL  DRAWING. 

Draw  all  necessary  lines  in  light  pencil,  then  submit  the  drawing 
for  inspection.  In  inking,  ink  only  those  lines  shown  on  the 
plate ;  give  all  dimensions — supplying  those  marked  X — and  finish 
the  sheet  by  lettering  it  as  shown. 

81.  PROBLEM  3: 

To  locate,  and  to  find  the  length  of  guy-wires  jor  a  smoke- 
stack. 

The  principles  of  geometry  involved  in  the  solution  of  this 
problem  are,  "To  find  the  point  in  which  a  given  line  pierces  a 
given  plane,  and  to  find  the  true  length  of  a  line." 

Let  the  problem  be  that  presented  by  Plate  5.  A  power- 
house of  the  shape — roof  half  pitch  (45°) — and  size  given,  is  to  have 
a  42"°  stack,  guyed  by  six  guy-wires,  arranged  as  shown,  and 
making  an  angle  of  45°  with  the  stack;  the  conditions  are  such 
that  two  of  the  guys  will  strike  the  roof  plane,  and  let  it  be  re- 
quired to  locate  the  ground  end  of  each  wire,  and  to  find  the 
length  of  each. 

A  cable  suspended  as  in  this  example  would  not  assume 
a  straight  line  as  shown,  but  would  assume  a  curve,  however  for 
the  problem  the  hypothetical  case  of  the  straight  line  is  to  be 
taken. 

To  find  the  points  in  which  the  guys  pierce  the  ground, 
revolve  one  of  them  parallel  to  the  vertical  plane  and  note  the 
distance  of  the  point  in  which  the  vertical  projection  of  the  guy 
strikes  the  ground -line  from  the  point  in  which  the  vertical  pro- 
jection of  the  center  line  of  the  stack  intersects  the  ground -line; 
with  this  length  as  a  radius  and  the  horizontal  projection  of  the 
center  line  of  the  stack — the  center  of  the  circle — as  a  center, 
describe  a  circle  intersecting  the  horizontal  projections  of  the 
guys ;  these  points  of  intersection  will  be  the  horizontal  projection 
of  the  points  in  which  the  guys  pierce  the  ground, — they  may  be 
located  by  referring  them  to  the  foundation  of  the  building. 
The  true  length  of  the  guys  reaching  the  ground,  is,  evidently, 
he  length  of  the  45°  line  on  the  vertical  plane — the  vertical  pro- 
tection of  a  guy  when  parallel  to  F. 


PRACTICAL   PROBLEMS. 


127 


PLATE  No.  5. 


128  ADVANCED  MECHANICAL  DRAWING. 

To  find  the  point  in  which  the  guy  on  the  right  pierces  the 
roof  plane  (the  example  on  the  left  is  a  case  of  simple  projection, 
the  plane  of  the  roof,  there,  being  perpendicular  to  F),  note  that 
the  plane  of  the  roof  strikes  the  ground  B  distance  from  the  founda- 
tion, which  enables  one  to  draw  the  trace  of  the  roof  plane  (t-T-f), 
and  with  the  projections  of  the  guy  given,  to  find  the  point  in  which 
the  guy  pierces  the  plane  T.  The  true  length  of  the  guys  are 
found  by  revloving  them  into  parallelism  with  F. 

Directions  for  Drawing. 

Execute  a  scale  drawing  (TV"  =  i')  according  to  the  dimen- 
sionsgiven,  drawing  all  necessary  lines  in  light  pencil,  then  submit 
the  drawing  for  inspection.  In  inking,  omit  all  construction 
lines,  give  all  dimensions — supplying  the  required  lengths — and 
finish  the  sheet  by  lettering  it  as  shown. 

82.  PROBLEM  4: 

To  find  the  shape,  size,  and  bevels  of  the  three  pieces  forming 
the  triangular  object  shown  in  Plate  6,  and  to  find  the  angles  be- 
tween the  pieces. 

This  problem  is  a  typical  one,  and  involves  the  principles 
of  geometry,  "To  find  the  true  length  of  a  line,  and  to  find  the 
true  size  of  an  angle."  It  is  met  with  in  practice  in  many  different 
forms. 

Sufficient  dimensions  are  given  that  the  section  of  each  piece 
may  be  found,  then  each  piece  revolved  about  one  of  its  edges 
into  parallelism  with  the  horizontal  plane — a  position  from  which 
the  bevels  may  be  found. 

Directions  for  Drawing. 

Execute  a  full-sized  drawing  according  to  the  dimensions 
given,  drawing  the  plan  drawing  first,  then  the  elevation,  then 
draw  the  sections;  taking  one  piece  at  a  time,  assume  it  to  be 
removed  to  one  side,  then  revolve  it  into  parallelism  with  H; 
scale  this  drawing  and  supply  the  dimensions  marked  X.  To 
find  the  angles  D,  E,  F,  G,  H,  and  /,  pass  a  horizontal  pro- 
jecting plane  perpendicular  to  one  side  of  the  bevel ;  this  will  cut 


PRACTICAL  PROBLEMS. 


129 


PLATE   No.  6. 


130  ADVANCED  MECHANICAL  DRAWING. 

the  required  angle,  which  may  be  shown  by  revolving  the  plane 
parallel  to  H.  The  angles  between  the  pieces  are  found  by 
adding  the  bevels  of  each  piece,  as  shown  by  the  plate. 

In  addition  to  the  above  statement,  let  it  be  required  to 
draw  a  half-size  isometric  drawing  of  the  object,  taking  the 
dimensions  from  the  mechanical  drawing  by  scaling  it.  This 
is  done  in  accordance  with  the  principles  of  isometric  drawing 
as  given  in  Chapter  I. 

Draw  all  necessary  lines  in  light  pencil,  then  submit  the 
drawing  for  inspection.  In  inking,  ink  only  those  lines  shown 
on  the  plate;  give  all  dimensions — supplying  those  marked  X — 
and  finish  the  sheet  by  lettering  it  as  shown. 


Plate  7,  showing  a  skeleton  drawing  of  a  type  of  locomo- 
tive, illustrates  a  number  of  typical  problems  in  intersections 
and  developments  met  with  in  sheet-metal  work.  Six  plates — 
those  numbered — are  chosen  for  example,  and  a  mechanical 
drawing  of  each  is  given  at  the  bottom  of  the  plate. 


PLATE  No.  7. 


PRACTICAL  PROBLEMS. 


132  ADVANCED  MECHANICAL  DRAWING 


83.  PROBLEM  5: 

To  show  the  layout  for  the  sheets  forming  a  locomotive 
stack.  Plate  8  (see  Plate  7,  also). 

An  analysis  of  the  stack  shows  it  to  be  made  up  of  a  right 
cylinder  (No.  3)  and  parts  of  two  right  cones  (Nos.  i  and  2); 
therefore,  produce  the  sides  of  the  conical  parts  to  complete  the 
cones,  and  then  lay  out  the  developments  as  suggested  by  the 
lines  of  the  plate. 

Directions  for  Drawing. 

Execute  a  scale  drawing  (|"  =  i')  according  to  the  di- 
mensions given,  drawing  all  necessary  lines  in  light  pencil, 
then  submit  the  drawing  for  inspection.  In  inking,  omit  all 
construction  lines;  give  all  dimensions — supplying  those  marked 
X — and  finish  the  sheet  by  lettering  it  as  shown. 


PRACTICAL   PROBLEMS. 


'33 


PLATE  No.  8. 


00 
O 

o: 
CL 

_j 
o 

i 


SlN3kNd013A3Q 


<-r -9,* r> 

k— 1—7,5  .e-j—^j 

rrj 


//^T*^!  ®  *•  *  *       77^2       H 


134  ADVANCED  MECHANICAL  DRAWING. 


84.  PROBLEM  6: 

To  lay  out  the  sheet  for  the  smoke-box  (No.  4}  and  the 
second  ring  (No.  5)  of  the  barrel  of  a  locomotive.  Plate  9  (see 
Plate  7,  also). 

An  inspection  of  the  plate  shows  these  two  sheets  to  be 
right  cylinders,  in  the  case  of  No.  5  intersecting  another  right 
cylinder  (the  sand-dome),  and  in  No.  4  intersecting  a  right  cylin- 
der (the  stack)  and  a  rectangular  solid  (the  exhaust-nozzle  in 
the  smoke-box). 

To  develop  the  sheets,  find  the  above  intersections,  then 
proceed  as  is  suggested  by  the  lines  of  the  plate. 

Directions  for  Drawing. 

Execute  a  scale  drawing  ( J"  =  i')  according  to  the  dimen- 
sions given,  drawing  all  necessary  lines  in  light  pencil,  then 
submit  the  drawing  for  inspection.  In  inking,  omit  all  construc- 
tion lines;  give  all  dimensions — supplying  those  marked  X — and 
finish  the  sheet  by  lettering  it  as  shown. 


PRACTICAL  PROBLEMS. 


135 


PLATE  No.  9. 


I36  ADVANCED  MECHANICAL  DRAWING. 

85.  PROBLEM  7: 

To  lay  out  the  slope-sheet  (No.  6)  and  the  outside  sheet 
(No.  7)  oj  a  locomotive  boiler.  Plate  10  (see  Plate  7,  also). 

An  analysis  of  the  sheets  shows  that  No.  6,  the  slope-sheet, 
is  one  half  a  conoid  and  one  half  a  right  cylinder,  and  that  No.  7 
is  an  elliptical  right  cylinder. 

To  develop  sheet  No.  7  find  the  intersection  with  the 
steam-dome,  and  proceed  as  suggested  by  the  lines  of  the  plate. 

To  develop  sheet  6,  draw  a  number  of  elements  of  the 
conoid,  as  the  lines  1-9,  2-8,  etc.,  of  the  end  elevation  drawing, 
and  project  them  to  the  side  elevation;  next,  begin  at  the  center 
line  9-1  of  the  development,  and  lay  off  a  length  9-1  equal  to 
the  true  length  of  the  element  9-1  (taken  from  the  side  elevation); 
now  find  the  true  length  of  the  arc  1-2,  end  elevation,  and  with 
this  length  as  a  radius  and  the  point  i  of  the  developed  element 
9-1  as  a  center,  strike  an  arc ;  now  find  the  true  length  of  the  diag- 
onal line  9-2  (end  elevation) — the  line  9-2'  of  the  side  elevation — 
and  with  this  length  as  a  radius  and  a  center  at  the  point  9  of 
the  developed  element  9-1  strike  a  second  arc  intersecting  the 
first  arc;  this  will  give  the  point  2  of  the  development,  and  com- 
plete the  development  of  the  triangle  9-1-2 — a  small  portion  of 
the  surface.  Proceed  in  this  manner  (dividing  the  surface  into 
a  number  of  small  sections),  and  lay  the  several  sections  out 
with  the  common  side  of  each  two  sections — the  diagonal  lines 
as  the  line  9-2' — common  to  the  developed  two  sections,  and 
draw  a  curved  line  through  the  corner-points  of  the  sections; 
as  shown,  and  a  figure,  M-M-M-M,  will  be  obtained,  which 
represents  the  development  of  the  conoid  part  of  the  sheet.  The 
development  of  the  remainder  of  the  sheet  is  that  of  a  right  cylin- 
der, and  the  procedure  is  clearly  indicated  by  the  plate. 
Directions  for  Drawing. 

Execute  a  scale  drawing  (i"  =  i')  according  to  the  dimen- 
sions given,  drawing  all  necessary  lines  in  light  pencil,  then 
submit  the  drawing  for  inspection.  In  inking,  omit  all  con- 
struction lines,  give  all  dimensions — supplying  those  marked 
X — and  finish  the  sheet  by  lettering  it  as  shown. 


PRACTICAL   PROBLEMS. 


137 


PLATE  No.  10. 


1 38  ADVANCED  MECHANICAL  DRAWING. 

86.  PROBLEM  8: 

To  find  the  shape  and  size  of  the  plates  used  to  form  an  elbow. 

Let  the  problem  be  that  presented  by  Plate  n,  and  let 
it  be  required  to  lay  out  templets  for  the  three  plates  forming 
the  elbow,  the  allowances  for  lap  at  the  joints  being  disregarded. 

An  inspection  of  the  elbow  shows  it  to  be  made  up  of  three 
figures,  "4"  and  "C"  being  right  cylinders  of  circular  section, 
and  "B"  an  oblique  cone,  and  the  principles  of  geometry  involved 
in  the  solution  of  the  problem  to  be  the  development  of  these 
three  figures,  as  a  drawing  of  the  developments  executed  on  heavy 
paper,  cut  out  and  duplicated  in  wood  or  thin  sheet  metal,  or 
the  paper  laid  directly  on  a  sheet  of  metal  for  laying  out  the 
plates,  would  be  called  a  set  of  templets  for  the  job. 

To  develop  the  plates,  first  pass  a  number  of  intersecting 
planes  intersecting  the  three  figures  in  elements  (for  convenience, 
it  is  suggested  that  these  planes  be  made  to  divide  the  bases  into 
an  equal  number  of  equal  arcs,  twelve  being  a  good  working 
number);  then,  beginning  with  cylinder  ".4,"  lay  off  the  line 
X-Y-X  equal  to  the  circumference  of  "4"  and  erect  the  per- 
pendiculars representing  the  elements  cut  by  the  intersecting 
planes  (these  lengths  are  taken  directly  from  the  elevation  draw- 
ing of  "-4"),  and  through  their  extremes  draw  the  curve  /-/ 
representing  the  line  of  the  upper  base  of  the  cylinder. 

To  develop  the  cone  u£,"  the  true  length  of  each  element 
must  first  be  obtained  by  revolving  it  into  parallelism  with  the  hori- 
zontal plane  (this  is  a  third  angle  projection)  and  with  the  length 
of  the  lower  base  of  the  cone  known — for,  since  the  cone  is  fitted 
to  cylinder  "^4,"  the  circumference  of  the  bases  are  equal — 
proceed  to  construct  development  "B"  as  follows*  Select  a 
center- point,  C,  and  draw  the  center  line  C-G-"j  equal  in  length 
to  the  true  length  of  element  j-C.  (It  should  be  noted  that  the 
figures  are  cut  along  the  outside  element — elevation  drawing — 
cut  by  the  intersecting  plane  i-C  of  the  plan  drawing.)  With  C 
as  a  center  and  a  radius  equal  to  the  true  length  of  element  6-C 
describe  an  arc;  then,  with  point  7  as  a  center,  and  a  radius  k, 
taken  from  the  development  of  cylinder  "A"  describe  an  arc 


PRACTICAL   PROBLEMS. 


139 


PLATE  No.  ii. 


140  ADVANCED  MECHANICAL  DRAWING. 

intersecting  the  first  arc  —  the  point  of  intersection,  e,  will  be 
the  locus  of  the  lower  base  end  of  element  6-C;  this  element  may 
then  be  drawn  by  connecting  points  e  and  C;  similarly  combining 
the  true  length  of  each  element  with  the  proper  distance  between 
elements,  taken  from  development  "4,"  obtain  a  series  of  points 
y-e-d-c,  etc.,  through  which  the  line  1-7-1,  representing  the  de- 
veloped circumference  of  the  lower  base  of  the  cone,  is  drawn. 
With  these  points,  and  the  center-point  C,  known,  it  is  a  simple 
procedure  to  lay  off  the  true  length  of  each  element  and  through 
the  extremes  to  draw  the  developed  line  of  the  upper  base,  and 
thus  complete  the  development. 

Fig.  "C"  is  a  right  cylinder  neither  base  of  which  is  at 
right  angles  to  the  elements.  To  develop  this  cylinder,  assume 
an  intermediate  base,  m-n,  the  plane  of  which  is  perpendicular 
to  the  elements  and  which  will  develop  as  the  straight  line  m-n] 
this  line  is  used  as  a  base  line  for  drawing  the  development,  the 
method  of  procedure  being  similar  to  that  used  for  developing 
cylinder  "A." 

Directions  for  Drawing. 

Execute  a  scale  drawing  (ii"  =  i')  according  to  the  dimen- 
sions given,  drawing  all  necessary  lines  in  light  pencil,  then 
submit  the  drawing  for  inspection.  In  inking,  omit  all  construc- 
tion lines,  give  all  dimensions,  and  finish  the  sheet  by  lettering 
it  as  shown. 

87.  PROBLEM  9: 

To  lay  out  a  reducing  breeching  for  a  hot-air  conduit  in  which 
the  main  is  26"  in  diameter,  reducing  in  one  leg  to  20"  in  diameter, 
and  in  the  other  to  14''  in  diameter;  the  angle  between  the  legs 
and  the  main  to  be  135°. 

Let  the  problem  be  that  presented  by  Plate  12  and  illus- 
trated by  Fig.  109,  and  let  it  be  required  to  develop  sheets  A 


To  lay  out  the  breeching,  first  draw  the  center  lines  making 
the  required  angles  with  one  another,  then  draw  the  end  view 
of  the  26"  main;  next,  at  the  intersection  of  the  center  lines. 


PRACTICAL   PROBLEMS 


141 


PLATE  No.  12. 


142 


ADVANCED  MECHANICAL  DRAWING. 


PLATE  No.  12  A. 


OF  THE 

UNIVERSITY 

OF 


PRACTICAL   PROBLEMS 


'43 


or  axes,  point  Z>,  draw  bases  of  right  cones,  the  diameters  of 
which  are  a  little  greater  than  the  diameter  of  the  main  (in  this 
case  27"),  and  the  planes  of  which  are  perpendicular  to  the 
respective  axes  of  the  reducing  legs  of  the  breeching.  This  done, 
draw  lines  tangent  to  the  bases  of  the  cones  and  the  outline  of 
the  cylindrical  main  (end  view),  and  produce  them  to  an  in- 
tersection with  the  plane  of  the  axes — the  points  of  intersection 


FIG.  109. 

will  define  the  apices  of  the  cones  assumed;  next,  find  the  in- 
tersection of  the  cylindrical  main  with  the  conical  reducing  legs, 
and  the  intersection  of  the  legs  themsleves,  then  cut  the  legs  off 
at  the  proper  point  to  give  the  required  diameter. 

Sheets  A  and  B  forming  a  right  cylinder  and  a  right  cone, 
respectively,  the  developments  are  readily  obtained,  and  are  to 
be  laid  out  in  accordance  with  Plate  12,  A. 
Directions  for  Drawing. 

Execute  a  scale  drawing  (ij"  =  i')  according  to  the  dimen- 
sions given,  drawing  all  necessary  lines  in  light  pencil,  then 
submit  the  drawings  for  inspection.  In  inking,  ink  only  those 
lines  shown  on  the  Plates,  give  all  dimensions,  and  finish  the 
sheets  by  lettering  them  as  shown. 


144  ADVANCED  MECHANICAL  DRAWING. 

88.  PROBLEM  10: 

To  lay  out  the  plates  for  a  screw-grain  conveyor. 

Let  the  problem  be  that  presented  by  Plate  13,  which 
illustrates  a  portion  of  a  form  of  grain-conveyor  (small  model), 
and  let  it  be  required  to  lay  out  the  blade  in  the  most  economical 
manner  for  punching  from  sheet  metal. 

The  problem  is  typical  of  a  form  of  conveyor  of  a  wide 
range  of  usefulness;  post-hole  augers,  the  helical,  inclined  plane 
up  which  the  circus  performer  a-foot  of  a  ball  or  astride  a  wheel 
wends  his  way,  are  other  examples  of  the  surface.  For  punching, 
it  is  evident  that  by  laying  out  the  blade  in  two  sections  for  each 
convolution  one  can  effect  a  saving  of  material,  and,  disregarding 
the  question  of  lap  and  method  of  fastening  to  the  central  core 
— items  to  be  considered  in  actual  manufacture — let  it  be  re- 
quired to  develop  the  blade. 

Inspecting  the  figure  one  recognizes  a  practical  applica- 
tion of  the  right  helicoid,  and  the  principle  of  geometry  involved 
in  the  solution  of  the  problem  is,  "The  development  of  a  right 
helicoid."  This  being  a  surface  of  double  curvature — a  warped 
surface — theoretically  it  cannot  be  developed,  though  practically 
it  can  be  very  closely  approximated. 

To  develop  the  figure,  draw  the  straight  line  5-5  equal 
to  the  true  length  of  an  element — it  is  evident  that  all  of  the 
elements  are  of  the  same  length — then,  find  the  true  distance 
between  elements,  inner  and  outer  ends — these  lengths  are  also 
uniform;  this  distance  is  the  true  length  of  the  cord  of  arc  B 
and  of  arc  ^4,  respectively.  With  these  lengths  as  radii,  and  the 
extremes  of  the  line  5-5  as  centers,  describe  arcs  as  shown.  Next, 
find  the  true  length  of  the  diagonal  C,  which  gives  the  last  of  the 
lengths  required  for  the  development;  CL  being  this  length, 
and  dimension  E  the  length  of  an  element,  the  various  lengths 
are  combined  as  shown,  and  the  points  4,  3,  2,  etc.,  obtained; 
these  points  form  the  locus  of  the  curves  forming  the  outline  of 
the  development. 

In  a  carefully  executed  drawing  these  curves  will  be  found 
to  be  concentric  circles,  and  by  using  any  three  points  of  either 


PRACTICAL  PROBLEMS. 


PLATE  No.  13. 


146  ADVANCED  MECHANICAL  DRAWING. 

extreme    the   center  of  the  circles  may  be  found  and  the  arcs 
drawn. 

Directions  for  Drawing. 

Execute  a  full-sized  drawing  according  to  the  dimensions 
given,  arranging  the  punchings  as  shown;  draw  all  necessary 
lines  in  light  pencil,  then  submit  the  drawing  for  inspection. 
In  inking,  omit  all  construction  lines,  give  all  dimensions — supply- 
ing dimensions  marked  X — and  finish  the  sheet  by  lettering  it  as 
shown. 

89.  PROBLEM  n: 

To  find  the  shape  and  size  oj  certain  plates  forming  part  of 
a  positive- feed  mechanism. 

In  Problem  10  is  found  a  practical  application  of  the  right 
helicoid;  the  oblique  helicoid  is  also  often  met  with  in  practice. 
It  is  obvious  that  the  sides  of  a  square  thread  are  right  helicoids,' 
the  sides  of  a  V  thread  are  examples  of  the  oblique  helicoid. 

Let  it  be  required  to  construct  a  V-threaded  screw  of  plates 
of  metal  to  form  a  positive  feed  for  some  such  mechanism  as 
the  housewife's  food-grinder — this  is  named  as  an  example 
because  of  its  familiarity,  though  in  this  apparatus  the  "screw" 
is  usually  of  cast  iron.  The  more  usual  application  of  the  oblique 
helicoid  is  in  certain  forms  of  screw-propellers  for  boats,  vanes 
for  water-wheels,  in  positive  pressure  blowers,  etc. 

Assuming  that  if  one  side  of  the  screw-thread  can  be  laid 
out,  the  other  side  may  be  readily  obtained,  the  problem  deals 
with  a  single  face  of  the  thread  and  is  presented  by  Plate  14 
illustrating  one  convolution  of  this  face,  the  problem  being  to 
develop  the  plate  shown. 

The  figure  is  a  warped  surface,  hence  the  development 
can  only  be  approximated.  The  method  of  procedure  is  exactly 
as  given  for  developing  the  right  helicoid  of  Problem  10;  dimen- 
sions AL,  BL,  CL,  and  E  (see  development)  being  equal  to  the 
true  distance  between  elements — inner  and  outer  extremes — the 
true  length  of  a  diagonal,  and  the  true  length  of  an  element, 
respectively;  it  is  evident  that  these  dimensions  are  uniform 


PRACTICAL   PROBLEMS. 


147 


PLATE  No.  14. 


148  ADVANCED  MECHANICAL  DRAWING. 

throughout  the  development,  and  when  laid  out  will  give  limiting 
curves,  which  are  circular  arcs. 

Directions  for  Drawing. 

Execute  a  full-sized  drawing  according  to  the  dimensions 
given,  drawing  all  necessary  lines  in  light  pencil,  then  submit 
the  drawing  for  inspection.  In  inking,  ink  only  those  lines  shown 
on  the  plate,  give  all  dimensions,  supplying  dimensions  marked 
X — these  would  be  necessary  for  laying  out  on  a  sheet  of  metal 
— and  finish  the  sheet  by  lettering  it  as  shown. 

90.  PROBLEM  12: 

To  find  the  angle  necessary  jor  the  section  of  an  angle-iron 
jor  framing  the  corners  of  a  metal  coal-hopper. 

Let  the  problem  be  that  presented  by  Plate  15,  an  in- 
spection of  which  shows  the  hopper  to  be  formed  by  four  inclined 
planes,  and  the  principle  of  geometry  involved  to  be,  "To  find 
the  angle  between  two  planes." 

To  find  the  required  angle  pass  an  auxiliary  plane  per- 
pendicular to  the  intersection  of  the  side  planes  of  the  hopper, 
and  find  the  line  cut  from  each  side  plane  by  the  auxiliary  plane 
— the  angle  between  these  two  lines  will  be  the  required  angle. 
The  construction  is  clearly  shown  on  the  plate. 

Directions  for  Drawing. 

Execute  a  scale  drawing  (£"  =  1')  according  to  the  dimen- 
sions given,  drawing  all  necessary  lines  in  light  pencil,  then 
submit  the  drawing  for  inspection.  In  inking,  ink  only  those  lines 
shown  on  the  plate,  give  all  dimensions — supplying  the  angle 
A — and  finish  the  sheet  by  lettering  it  as  shown. 


PRACTICAL   PROBLEMS. 


149 


PLATE   No.   is. 


ADVANCED  MECHANICAL  DRAWING. 


SHADOWS. 

91-93.  PROBLEMS   13,  14,  and  15: 

To  find  some  elementary  shadows. 

Let  the  problems  be  those  presented  by  Plates  16,  17, 
and  1 8,  respectively,  and  let  it  be  required  to  find  all  of  the  shadows 
cast  by  the  figures.  The  principle  of  Descriptive  Geometry 
involved  in  these  problems  is,  "To  find  the  point  in  which  a 
line  pierces  the  planes  of  projection." 

Before  attempting  these  exercises  the  student  should  read 
Chapter  II,  and  should  then  study  the  figures  given,  and  select 
those  points  and  lines  which  cast  the  limiting  lines  of  the  shadow, 
and  thus  find  the  shadows  with  as  few  working  lines  as  possible. 

In  Problem  15,  Plate  18,  in  addition  to  finding  the  shadow, 
let  it  be  required  to  illustrate  the  problem  with  an  isometric 
drawing  of  it. 

Directions  for  Drawing. 

Execute  a  full-sized  drawing  of  each  problem  (three  sepa- 
rate exercises)  according  to  the  dimensions  given,  drawing  all 
necessary  lines  in  light  pencil,  then  submit  the  drawing  for 
nspection.  'In  inking,  ink  only  those  lines  shown  on  the  plate, 
and  the  outline  of  the  shadow,  rule  the  shadow  in  as  shown 
by  the  examples  of  Chapter  II,  omit  all  dimensioning,  and  finish 
the  sheet  by  lettering  it  as  shown. 


PRACTICAL   PROBLEMS. 


PLATE   No.   16. 


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ADVANCED  MECHANICAL   DRAWING. 


PLATE  No.  17. 


PRACTICAL  PROBLEMS. 


'53 


PLATE  No.  18. 


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154  ADVANCED  MECHANICAL   DRAWING. 

94.  PROBLEM  16: 

To  find  the  shadow  cast  by  a  taboret. 

Let  the  problem  be  that  presented  by  Plate  19,  and  let 
it  be  required  to  find  all  of  the  shadows  cast  by  the  object. 

Directions  for  Drawing. 

Execute  a  scale  drawing  to  a  scale  of  2"  =  i',  drawing 
all  necessary  lines  in  light  pencil,  then  submit  the  drawing  for 
inspection.  In  inking,  ink  only  "those  lines  shown  on  the  plate, 
and  the  outline  of  the  shadow,  rule  in  the  shadow,  omit  all  dimen- 
sioning, and  finish  the  sheet  by  lettering  it  as  shown. 

95-96.  PROBLEMS  17  and  18: 

To  find  the  shadow  cast  on  a  double  curved  surface. 

Problem  17: 

Find  the  shadow  of  the  niche  shown  on  page  52  by 
either  assuming  dimensions  and  drawing  a  similar  figure,  or 
by  enlarging  the  cut ;  in  either  case,  draw  a  figure  that  will  nearly 
fill  the  sheet,  reserving,  however,  a  space  for  the  title  and  signa- 
ture. 

Problem  18: 

Find  the  shadow  of  a  sphere. 

Let  the  sphere  be  2j"  in  diameter;  let  it  be  equidistant 
— say  if" — from  both  of  the  planes  of  projection;  let  the  ground- 
line  be  3!"  above  the  lower  border  line  of  the  sheet  and  let  the 
center  of  the  sphere  be  4^"  in  from  the  left  border  line,  and  let 
it  be  required  to  find  the  shadow  on  the  sphere  and  on  the  planes 
of  projection. 

Directions  for  Drawing. 

Each  problem  is  to  constitute  an  exercise  and  is  to  occupy 
an  entire  sheet.  Construct  the  drawings  in  accordance  with  the 
instructions  given,  find  the  shadows,  then  submit  the  sheets 
(one  at  a  time)  for  inspection.  In  inking,  ink  only  the  out- 
line of  the  figures  and  shadows,  rule  in  the  shadow,  omit  all 
dimensioning,  and  finish  the  sheets  by  lettering  them  in  accord- 
ance with  Plate  19. 


PRACTICAL  PROBLEMS. 


'55 


PLATE  No.  19. 


\< ,91— 


TO: 

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s^\ 


156  ADVANCED  MECHANICAL   DRAWING. 

PERSPECTIVE. 

97.  PROBLEM  19: 

To  find  some  elementary  perspectives. 

Let  the  problem  be  that  presented  by  Plate  20,  showing 
the  mechanical  drawings  for  a  hollow  cube,  a  hollow,  hexagonal 
prism,  and  a  hollow  cylinder,  also,  the  conditions  for  the  per- 
spective, and  the  required  perspectives,  and  let  it  also  be  re- 
quired to  show  all  of  the  shadowy  cast  by  the  objects. 

Before  attempting  the  execution  of  the  problem,  the  student 
should  read  Chapter  III. 

Directions  for  Drawing. 

Execute  a  full-sized  mechanical  drawing  of  the  objects, 
according  to  the  dimensions  given,  on  a  sheet  of  paper  other 
than  that  to  receive  the  perspective — the  finished  sheet — then 
cut  the  paper,  separating  the  plans  from  the  elevations,  and 
arrange  them  about  the  sheet  to  receive  the  perspective — the 
field  of  the  picture — as  shown;  find  the  perspectives,  then 
submit  the  drawing  for  inspection.  In  inking,  ink  only  the 
perspectives,  and  the  8"Xn"  border  line  of  the  sheet,  and 
finish  the  sheet  by  lettering  the  title  and  signature  only. 


PRACTICAL   PROBLEMS. 


157 


PLATE  No.  20. 


158  ADVANCED  MECHANICAL  DRAWING. 

98.   PROBLEM  20: 

To  find  the  perspective  of  a  flight  oj  stone  steps. 

Let  the  problem  be  that  presented  by  Plate  21,  showing 
the  plan  and  elevation  of  the  steps,  the  conditions  for  the  per- 
spective, and  the  required  perspective,  and  let  it  also  be  required 
to  show  all  of  the  shadows  cast. 

The  conditions  given  are  intended  more  as  an  example 
than  as  specific  instruction,  and  it  is  suggested  that  the  student 
assume  other,  similar  conditions. 

Directions  for  Drawing. 

Execute  a  mechanical  drawing  of  the  steps  to  a  scale  of 
i"  =  i'  according  to  the  dimensions  given,  on  a  sheet  of  paper 
other  than  the  sheet  to  receive  the  perspective,  then  cut  the  paper, 
separating  the  plan  and  elevation,  and  arrange  these  views  and 
the  sheet  to  receive  the  perspective  as  suggested  by  the  plate; 
find  the  perspective,  then  submit  the  sheet  for  inspection.  In 
inking,  ink  the  perspective  and  the  8"Xn"  border  line  only, 
and  finish  the  sheet  by  lettering  it  as  follows : 

TITLE, 

PERSPECTIVE. 

Practical  Problem  No.  2O. 

NAME  AND  DATE,  to  be  in  the  lower  right-hand  corner  of 
the  sheet 


PRACTICAL   PROBLEMS. 


'59 


PLATE  No.  21. 


160  ADVANCED  MECHANICAL   DRAWING. 

99.  PROBLEM  21 : 

To  find  the  perspective  oj  a  small  railway  station-house. 
Let  the  problem  be  that  presented  by  Plate  22,  illustrating 
a  set  of  conditions  (these  or  other  conditions  may  be  used). 

Directions  for  Drawing. 

Execute  a  mechanical  drawing  of  the  building  to  a  scale  of 
J"  =  i'  on  a  sheet  of  paper  other  than  the  sheet  to  receive  the 
picture,  then  cut  the  paper,  separating  the  views,  and  arrange 
them  about  the  field  of  the  picture  as  suggested  by  the  plate; 
find  the  perspective,  then  submit  the  sheet  for  inspection.  In 
inking,  ink  the  lines  of  the  perspective  and  the  border  line  only, 
and  finish  the  sheet  by  lettering  it  as  follows: 

TITLE, 

PERSPECTIVE. 

Practical  Problem  No.  21. 

NAME  AND  DATE,  to  be  in  the  lower  right-hand  corner  of  the 
sheet. 


PRACTICAL  PROBLEMS. 


161 


PLATE  No.  22. 


162 


ADVANCED  MECHANICAL  DRAWING. 


PRACTICAL  PROBLEMS. 


163 


100.   PROBLEM  22: 

To  find  the  perspective  0}  a  railway  arch. 

Let  the  problem  be  that  presented  by  Fig.  no,  and  illus- 
trated by  Fig.  in,  and  let  the  student  assume  his  own  conditions, 
such  that  the  perspective  will  look  well  on  a  standard  8"Xii" 
sheet  of  paper. 

Directions  for  Drawing. 

Execute  the  mechanical  drawings  of  the  arch  by  enlarging 
the  copy,  say  two  times;  then  separate  the  views  and  arrange 


FIG.  in. 

them  about  the  sheet  to  receive  the  picture  in  accordance  with 
the  conditions  assumed,  then  submit  the  arrangement  for  ap- 
proval; next,  find  the  perspective,  then  submit  the  drawing  for 
inspection.  In  inking,  ink  the  lines  of  the  perspective  and  the 
border  line  only,  and  finish  the  sheet  by  lettering  it  as  follows: 
TITLE, 

PERSPECTIVE. 

Practical  Problem  No.  22. 

NAME  AND  DATE,  to  be  in  the  lower  right-hand  corner  of  the 
sheet. 


1 64  ADVANCED  MECHANICAL  DRAWING. 

10 1.  PROBLEM  23: 

To  find  the  perspective  oj  an  architectural  arch. 
Let    the    problem    be    that    presented    by    Fig.    112    (the 
mechanical  drawings),  and  illustrated  by  Fig.  113,  and  let  the 
student  assume  his   own  conditions,   such  that   the  perspective 
will  look  well  on  a  standard  8"Xn"  sheet  of  paper. 

Directions  for  Drawing. 

Execute  the  mechanical  drawings  of  the  arch  by  enlarging 
the  copy,  say  two  times;  then  separate  the  views  and  arrange 
them  about  the  sheet  to  receive  the  picture  in  accordance  with 
the  conditions  assumed,  then  submit  the  arrangement  for  ap- 
proval; next,  find  the  perspective,  then  submit  the  drawing  for 
inspection.  In  inking,  ink  the  lines  of  the  perspective  and  the 
border  line  only,  and  finish  the  sheet  by  lettering  it  as  follows: 
TITLE, 

PERSPECTIVE. 

Practical  Problem  No.  23. 

NAME  AND  DATE,  to  be  in  the  lower  right-hand  corner  of 
the  sheet. 


PRACTICAL   PROBLEMS. 


165 


SIDE  LLEVATION 


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FIG.  us. 


FIG.  in. 


1 66  ADVANCED  MECHANICAL  DRAWING. 


FIRST  FLOOR  PLAN. 
FIG.  114,  A. 


PRACTICAL   PROBLEMS. 


167 


102.   PROBLEM   24. 

To  find  the  perspective  of  a  small  dwelling-Jtouse. 
Let  the  drawings  of  the  house  be  those  presented  by  Fig. 
114,   A,  Bj  C,  D,  and  E,  and  the  problem  that  illustrated  by 


FRONT  ELEVATION. 
FIG.  114,  B. 

Plate  23,  and  let  the  student  assume  his  own  conditions,  such 
that  the  perspective  will  look  well  on  a  standard  8"Xn"  sheet 
of  paper. 

Directions  for  Drawing. 

Execute  the  mechanical  drawings  of  the  house  (a  plan  and 
left  elevation  is  sufficient)  by  enlarging  the  copy,  say  two  times; 
then  separate  the  views  and  arrange  them  about  the  sheet  to 
receive  the  picture  in  accordance  with  the  conditions  assumed; 
then  submit  the  arrangement  for  approval;  next,  find  the  per- 
spective, then  submit  the  drawing  for  inspection.  In  inking, 


168 


ADVANCED  MECHANICAL  DRAWING. 


PLATE  No.  23. 


O  CD 

UJ  O 

CL  cr 

CO  a. 


Q. 


PRACTICAL  PROBLEMS. 


169 


170 


ADVANCED  MECHANICAL  DRAWING 


ink  the  lines  of  the  perspective  and  the  border  line  only,  and 
finish  the  sheet  by  lettering  it  as  shown. 


REAR  ELEVATION, 
FIG.  114,  D, 


PRACTICAL   PROBLEMS. 


I72  ADVANCED  MECHANICAL  DRAWING. 

103.   PROBLEM  25: 

To  find  the  "isometric  perspective"  of  a  house. 

Let  the  drawings  of  the  house  be  those  presented  by 
Fig.  114,  A,  Bj  C,  D,  and  E,  and  let  it  be  required  to  picture  the 
right  and  rear  elevations  in  isometric.  (See  Plate  24.) 

Directions  for  Drawing. 

Let  the  dimensions  for  the  drawing  be  obtained  by  scaling 
the  copy,  and  then  construct  the  drawing  one  and  one  half  times 
the  size  of  the  copy.  In  accordance,  then,  with  these  conditions 
and  the  principles  of  isometric  drawing  as  set  forth  in  Chapter 
I,  execute  the  drawing,  then  submit  it  for  inspection.  In  inking, 
ink  only  the  lines  shown  on  the  plate,  and  finish  the  sheet  by 
lettering  it  as  shown. 


PRACTICAL  PROBLEMS. 


PLATE  No.  24. 


174  ADVANCED  MECHANICAL  DRAWING. 


SUPPLEMENTAL. 
EXERCISES  IN  FREE-HAND  LETTERING. 

104.  Explanatory. — Since  there  is  so  little  free-hand  work — let- 
tering and  dimensioning — on  the  Plates  of  the  Course,  the  student 
is  apt  to  permit  his  hand  to  lose  the  dexterity  acquired  in  the 
execution  of  the  Plates  of  the  Course  in  Elementary  Mechanical 
Drawing,  and  the  following  exercises  are  offered  as  affording 
examples  for  practice,  and  it  is  suggested  that  they  be  inserted 
at  regular  intervals  in  the  course. 

In  all  of  the  exercises  the  letters  are  to  be  executed,  free-hand, 
in  ink  directly,  with  top  and  bottom  guide-lines  in  pencil  as 
the  only  guide,  except,  in  cases  where  the  letters  are  to  balance 
with  reference  to  something,  as  with  the  border  line,  when  they 
should  be  first  pencilled  in  to  insure  balance,  then  inked  in.  The 
student  is  to  make  a  choice  of  the  size  of  letters  and  spacing  by 
comparison  with  the  copy.  All  of  the  sheets  are  to  be  the  standard, 
8"Xn"  border,  9^X1 2"  outside  dimension,  sheet  of  the  course. 

Plate  25  illustrates  some  types  of  simple,  plain,  easily  exe- 
cuted free-hand  letters;  Plate  26,  the  sometimes  practice  of 
tabulating  standard  information  for  use  in  the  shop,  and  Plate  27, 
a  cover-sheet  for  a  folio  for  a  set  of  drawings,  such  as  the  exer- 
cises of  this  Course  in  Drawing. 


PRACTICAL   PROBLEMS. 


'75 


PLATE  No.  25. 


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PRACTICAL   PROBLEMS. 


177 


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JOHN   WILEY    &    SONS, 

NEW  YORK. 
LONDON:   CHAPMAN  &  HALL,  LIMITED. 

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AGRICULTURE. 
Armsby's  Manual  of  Cattle-feeding  ...............................  12010,  Si  75 

Principles  of  Animal  Nutrition  ........................  .  .8vo,    4  oo 

Budd  and  Hansen's  American  Horticultural  Manual: 


Part  I.  —  Propagation,  Culture,  and  Improvement  ................  i2mo, 

Part  n.—  Systematic  Pomology  ...............................  iamo, 

Downing's  Fruits  and  Fruit-trees  of  America  .........................  8vo, 

Elliott's  Engineering  for  Land  Drainage  ...........................  i2mo, 


Practical  Farm  Drainage 


Green's  Principles  of  American  Forestry  ........  .  ..................  i2mo, 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (WolL)  ...........  rzmo, 

Kemp's  Landscape  Gardening  ....................................  12010, 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration  .....  i2mo, 

Sanderson's  Insects  Injurious  to  Staple  Crops  .......................  i2mo, 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 

Insects  Injuring  Fruits.     (In  preparation.') 

Stockbridge's  Rocks  and  Soils  ......................................  8vo,  50 

Woll  s  Handbook  for  Farmers  and  Dairymen  .......................  i6mo.  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings  ............................  i2mo,  2  50 

Berg's  Buildings  and  Structures  of  American  Railroads  .................  4to,  5  oo 

Birkmire'-s  Planning  and  Construction  of  American  Theatres  ...........  8vo,  3  oo 

Architectural  Iron  and  SteeL  ..................................  8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings.  ...............  8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings  ...............  8vo,  3  50 

Skeleton  Construction  in  Buildings  .............................  8vo,  3  oo 

Briggs's  Modern  American  School  Buildings  .........................  8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings  .....................  8vo,  4  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten  ............  8vo,  3  50 

Fireproofing  of  Steel  Buildings  ...............................  .  .  8vo,  2  50 

French  and  Ives's  Stereotomy  ......................  '  ................  8vo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection.  .....................  i6mo,  i  oo 

Theatre  Fires  and  Panics  ....................................  I2mo,  i  50 

Holly's  Carpenters'  and  Joiners'  Handbook  .........................  i8mo,  75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods  ..................  8vo,  2  oo 

1 


Kidder's  Architect's  and  Builder's  Pocket-book.  Rewritten  Edition.  i6mo,mor.,  5  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Non-metallic  Minerals:    Their  Occurrence  and  Uses 8vo,  4  oo 

Monckton's  Stair-building 4to,  4  oo 

Patton's  Practical  Treatise  on  Foundations <  .8vo,  5  oo 

Peabody's  Naval  Architecture 8vo,  7  50 

Richey's  Handbook  for  Superintendents  of  Construction.     (In  press.) 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish . .8vo,  3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo.  i  50 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  oo 

Towne's  Locks  and  Builders'  Hardware i8mo,  morocco,  3  oo 

Wait's  Engineering  and  Architectural  Jurisprudence .  8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo.  3  oo 

Wood's  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and  Steel..  .8vo,  4  oo 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 

Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,  i  25 

The  World's  Columbian  Exposition  of  1893 Large  4to,  i  oo 

ARMY  AND  NAVY. 
Bernadou's  Smokeless  Powder,  Nitro-cfcllulose,  and  the  Theory  of  the  Cellulose 

Molecule I2mo,  2  50 

*  Bruff's  Text-book  Ordnance  and  Gunnery 8vo,  6  oo 

Chase's  Screw  Propellers  and  Marine  Propulsion 8vo ,  3  oo 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squire's  Polarizing  Photo-chronograph 8vo,  3  oo 

Cronkhite's  Gunnery  for  Non-commissioned  Officers 241110.  morocco,  2  oo 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

Sheep,  7  5<> 

De  Brack's  Cavalry  Outpost  Duties.     (Carr.) 24010  morocco,  2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,    15  oo 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  oo 

*  Dyer's  Handbook  of  Light  Artillery I2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

*  Fiebeger's  Text-book  on  Field  Fortification Small  8vo,  2  oo 

Hamilton's  The  Gunner's  Catechism i8mo,  i  oo 

*  Hoff's  Elementary  Naval  Tactics 8vo,  i  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  4  oo 

*  Ballistic  Tables 8vo,  i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II . .  8vo.  each,  6  oo 

*  Mahan's  Permanent  Fortifications.     (Mercur.) 8vo,  half  morocco,  7  So 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  oo 

*  Elements  of  the  Art  of  War *. .  -8vo.  4  oo 

Metcalf 's  Cost  of  Manufactures— And  the  Administration  of  Workshops,  Public 

and  Private 8vo,  5  oo 

*  Ordnance  and  Gunnery.     2  vols. I2mo,  5  oo 

Murray's  Infantry  Drill  Regulations i8mo.  paper,  10 

Nixon's  Adjutants'  Manual 24mo,  i  oo 

Peabody's  Naval  Architecture 8vo,  7  5° 

2 


*  Phclps's  Practical  Marine  Surveying 8vo,    2  50 

Powell's  Army  Officer's  Examiner tamo.   4  oo 

Sharpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco,    i  50 

*  Walke's  Lectures  on  Explosives 8vo,    4  oo 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo. 

Winthrop's  Abridgment  of  Military  Law i2mo. 

Woodhull's  Notes  on  Military  Hygiene i6nio. 

Young's  Simple  Elements  of  Navigation i6mo  morocco, 


Second  Edition,  Enlarged  and  Revised i6mo,  morocco, 

ASSAYING. 


Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

lamo,  morocco,  i   50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments ....  8vo,  3  oo 

Miller's  Manual  of  Assaying izmo.  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ore* 8vo.  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  oo 

Dike's  Modern  Electrolytic  Copper  Refining 8vo.  oo 

Wilson's  Cyanide  Processes iamo,  50 

Chlorination  Process xamo,  50 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers. 870,  2  50 

Craig's  Azimuth 4to.  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo.  3  oo 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  a  50 

•  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

•  White's  Elements  of  Theoretical  and  Descriptive  Astronomy iamo.  2  oo 

BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,  i  25 

Thomt'  and  Bennett's  Structural  and  Physiological  Botany :6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo.  2  oo 

CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables iamo,  125 

Allen's  Tables  for  Iron  Analysis 8vo.  3  oo 

Arnold's  Compendium  of  Chemistry.     (MandeL) Small  8vo.  3  so 

Austen's  Notes  for  Chemical  Students iamo,  i  50 

*  Austen  and  Langworthy.      The   Occurrence   of  Aluminium   in  Vegetable 

Products,  Animal  Products,  and  Natural  Waters 8vo.  a  oo 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule xamo,  2  50 

Bolton's  Quantitative  Analysis 8vo,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  x  50 

Brush  and  Pen  field's  Manual  of  Determinative  Mineralogy 8vo.  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.  (Boltwood.)  . . .  8vo,  3  oo 

Conn's  Indicators  and  Test-papers. .  •. xamo.  2  oo 

Tests  and  Reagents 8vo.  3  oo 

Craft's  Short  Course  in  Qualitative  Chemical  Analysis.  (Schaeffer.) iamo,  x  50 

Dolezalek's  Theory   of   the   Lead   Accumulator   (Storage   Battery).    (Von 

Ende) iamo.  a  50 

Drechsel's  Chemical  Reactions.     (Merrill.) ...    iamo.  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rout's  Enzymes  and  their  Applications.     (Prescott. ) 8vo,  3  oo 

3 


Erdmann's  Introduction  to  Chemical  Preparations.     (Dumap.) i2mo,    i  25 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe 

i2mo,  morocco,    i  50 

Fowler's  Sewage  Works  Analyses izmo,    2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,    5  oo 

Manual  of  Qualitative  Chemical  Analysis.  Parti.  Descriptive.  (Wells.)  8vo,    3  oo 
System  of  Instruction   in    Quantitative   Chemical  Analysis.     (Cohn.) 

2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health I2mo,    i  50 

Furman's  Manual  of  Practical  Assaying 8vo,    3  oo 

Getman's  Exercises  in  Physical  Chemistry I2mo, 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,    i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     ( WolL) i2mo,    2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (MandeL) 8vo,   400 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,    i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,   2  50 

Hinds's  Inorganic  Chemistry 8vo,    3  oo 

•  Laboratory  Manual  for  Students i2mo,        75 

Holleman's  Text-beok  of  Inorganic  Chemistry.     (Cooper.) 8vo,    2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,    2  50 

•  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) X2mo,    i  oo 

Hopkins's  Oil-chemists'  Handbook 8vo,    3  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,     i  25 

Keep's  Cast  Iron 8vo,    2  50 

.Ladd's  Manual  of  Quantitative  Chemical  Analysis 12 mo,    i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,   3  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz.) i2mo,    i  oo 

Application  of  Some  General  Reactions    to    Investigations  in  Organic 

Chemistry.     (Tingle.) 12010,    i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,    7  50 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  t  Lorenz.)  i2mo,    i  oo 
Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments..  .  .8vo,    3  oo 

Lunge's  Techno-chemical  Analysis.     (Cohn.) i2mo,    i  oo 

Handel's  Handbook  for  Bio-chemical  Laboratory I2mo,    i  50 

•  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe . .  i2ino,       60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo.   4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological) 12 mo,    i  25 

Matthews's  The  Textile  Fibres 8vo,    3  50 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.;.  .i2mo,    i  oo 

Miller's  Manual  of  Assaying i2mo,    i  oo 

Mixter's  Elementary  Text-book  of  Chemistry i2mo,    i  50 

Morgan's  Outline  of  Theory  of  Solution  and  its  Results i2mo,    i  oo 

Elements  of  Physical  Chemistry i2mo,    2  oo 

Hone's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,    i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I. Large  8vo,   5  oo 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,    2  oo 

O'DriscolTs  Notes  on  the  Treatment  of  Gold  Ores 8vo,    2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo,    i  50 

•  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,        50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) I2mov    i  50 

Poole's  Calorific  Power  of  Fuels 8vo,    3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo.   i  25 

4 


•  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air  .Water,  and  Food  from  a  Sanitary  Standpoint.  8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science xamo  i  oo 

Cost  of  Food  a  Study  in  Dietaries xamo,  i  oo 

•  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. — 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Riggs's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Rostoski's  Serum  Diagnosis.  (Bolduan.) 12010,  i  oo 

Ruddiman's  Incompatibilities  in  Prescriptions. 8vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vot  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.  (Orndorff.). . .  .8vo.  2  50 

Schimpf's  Text-book  o!  Volumetric  Analysis xamo,  2  50 

Essentials  of  Volumetric  Analysis xamo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco*  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists. . i6mo,  morocco*  2  oo 

Stockbridge's  Rocks  and  Soils 8vo.  2  50 

•  Tollman's  Elementary  Lessons  in  Heat 8vo,  i  50 

•  Descriptive  General  Chemistry 8vo,  3  oo 

Treadwell's  Qualitative  Analysis.     (HalL) .*. 8vo,  3  oo 

Quantitative  Analysis.     (HalL) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies STO,  S  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

•  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Wassermann's  Immune  Sera:  Haemolysins,  Cytotoxins,  and  Precipitins.     (Bol- 
duan.)  i2mo,  i  oo 

Wells's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students 12010,  i  50 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wiechmann's  Sugar  Analysis Small  8vo.  2  50 

Wilson's  Cyanide  Processes. 12010,  i  50 

Chlorination  Process i amo,  i  50 

Wulling's  Elementary  Course  in  Inorganic  Pharmaceutical  and  Medical  Chem- 
istry  i2ino,  2  oo 

CIVIL  ENGINEERING. 
BRIDGES  AND    ROOFS.      HYDRAULICS.      MATERIALS   OP    ENGINEERING 

RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments I2mo(  3  oo 

Bixby's  Graphical  Computing  Table Paper  ioi  X  24}  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal    (Postage* 

27  cents  additional.) 8vo,  net*  3  50 

Comstock's  Field  Astronomy  for  Engineers. 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage lamo,  i  50 

Practical  Farm  Drainage xamo,  i  oo 

Folwell's  Sewerage.     (Designing  and  Maintenance.) STO.  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition  Rewritten 8vo .  3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements X2tnov  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

5 


Howe's  Retaining  Walls  for  Earth  ..„  ..............  ............         izmo,  i  25 

Johnson's  (j.  B.)  Theory  and  Practice  01  Surveying  .....  ........  Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods  ............  8vor  2  oo 

Lapiace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.)  ............  8^0.  s  oo 

•  Descriptive  Geometry  .........................................  8vo,  i  50 

Merriman'  s  Elements  of  Precise  Surveying  and  Geodesy.  .  ..............  8vo,  2  50 

Elements  of  Sanitary  Engineering  ...............................  8vo,  2  oo 

Merriman  and  Brooks's  Handbook  for  Surveyors  .............  i6mo,  morocco,  2  co 

Nugent's  Plane  Surveying  ..........................................  8vo  3  50 

Ogden's  Sewer  Design  ...........................................  Z2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering  ..........  ..........  8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching  .............  ............  4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage  ................  8vo,  350 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry  ................  8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.)  ..............  8vo,  2  50 

Sondericker's  Graphic   Statics,  with  Applications    to   Trusses,   Beams,   and 

Arches.  «.  ____  ...............................................  8vo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete  ..Plain  and  Reinforced.    (In  press.) 

•  Trautwine's  Civil  Engineer's  Pocket-book  ................  161110,  morocco,  5  oo 

Wait's  Engineering  and  Architectural  Jurisprudence  ...................  8vo,  6  oo 

Sheep,  6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 

tecture .................................................  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts  ............................................  8vo,  3  oo 

Warren's  Stereotomy  —  Problems  in  Stone-cutting  .....................  8vo,  2  50 

Webb's  Problems  in  the  U«e  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i   25 

•  Wheeler's  Elementary  Course  of  Civil  Engineering  ...................  8vo,  4  oo 

Wilson's  Topographic  Surveying  ...................................  8vo,  3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  ,8vo,  2  oo 

•  Thames  River  Bridge  ..................................  4to,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges  ........................  ;  ..............  8vo,  3  So 

Du  Bois's  Mechanics  of  Engineering.     VoL  H  .................  Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges  ..........................  4to,  5  oo 

Fowler's  Coffer-dam  Process  tor  Piers  .......................  .  .......  8vo,  2  50 

Ordinary  Foundations  .........................................  8vo,  3  50 


Greene's  Roof  Trusses  .............................................  8vo, 

Bridge  Trusses,  ........................................  ....  .8vo, 


Arches  in  Wood,  Iron,  and  Stone 


Howe's  Treatise  on  Arches  ........................................  8vo, 

Design  of  Simple  Roof  -trusses  in  Wood  and  Steel  ..................  8vo, 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures  .........................  Small  410,    10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.  —  Stresses  in  Simple  Trusses  ..............................  8vo,  2  50 

Part  IL—  Graphic  Statics  ......................................  8vo,  2  50 

Part  in  —Bridge  Design.    4th  Edition,  Rewritten  ................  8vo,  2  50 

Part  IV.—  Higher  Structures  ...................................  8vo,  2  50 

Horison's  Memphis  Bridge  .........................................  4to»   10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  .  i6mo.  morocco,  3  oo 

Specifications  for  Steel  Bridges  ................................  iamo,  i  25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs.Svo,  2  oo 

Wright's  Designing  of  Draw-spans: 

Part  L  —Plate-girder  Draws  ..................................  8vo.  2  50 

Part  II.—  Riveted-truss  and  Pin-connected  Long-span  Draws  .......  8vo,    2  50 

Two  parts  in  one  volume  ......................................  8vo,   3  SO 

6 


HYDRAULICS. 
Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from  an 

Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12 mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power 8vo,  5  oo 

Fuertes's  Water  and  Public  Health lamo.  i   50 

Water-filtration  Works xarno,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-work* 8vo ,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

Mason's   Water-supply.     (Considered   Principally   from   a   Sanitary   Stand- 
point)    3d  Edition,  Rewritten 8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics,     gth  Edition,  Rewritten 8vo,  5  oo 

•  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Schuyler's   Reservoirs  for  Irrigation.  Water-power,  and  Domestic  Water- 
supply Large  8vo,  5  oo 

•*  Thomas  and  Watt's  Improvement  of  Riyers.     (Post.,  44  c.  additional),  4to,  600 

Turneaure  and  Russell's  Public  Water-supplies. 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  oo 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4 to,  10  oo 

Weisbach's  Hydraulics  and  Hydraulic  Motor*.     (Du  Bois. ) 8vo,  5  oo 

Wilson's  Manual  of  Irrigation  Engineering Small  8vo.  4  oo 

Wolff's  Windmill  a*  a  Prime  Mover 8vo.  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 

MATERIALS  OP  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements. 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  4to,  5  oo 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edi- 
tion, Rewritten 8vo,  7  50 

Byrne's  Highway  Construction SFO,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     VoL  I Small  4to,  7  50 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Keep's  Cast  Iron N 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Maxtens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols, 8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials I2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richey's  Handbook  for  Building  Superintendents  of  Construction.     (7n  press.) 

Rockwell's  Roads  and  Pavements  in  France i2mo,  i  25 

7 


Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2mo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced.     (In 

press. ) 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  oo 

Part  I. — Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo 

Part  II. — Iron  and  Steel 8vo,  3  50 

Part  III. — A  Treatise  on  Brasses,  Bronzes,  and  Othsr  Alloys  and  their 

Constituents 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  oo 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Waddell's  De  Pontibus.     (A  Pocket-book  for  Bridge  Engineers.) .  .  i6mo,  mor.,  3  oo 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings :    Corrosion  and  Electrolysis  of  Iron  and 

Steel. 8vo.  4  oo 

RAILWAY  ENGINEERING. 

Andrews's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brooks's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butts's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.     i6mo,  morocco,  5  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:  (1879) Paper,  5  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills,  4to,.half  mor.,  25  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide ....  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  ot  Calculating  the  Cubic  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

1 2mo» morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction.     2d  Edition,  Rewritten i6mo,  morocco,  5  oo 

Wellington's  Economic  Theory*  of  the  Location  of  Railways Small  8vo,  5  oo 

DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "       Abridged  Ed 8vo,  i  50 

Coolidge's  Manual  ot  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to.  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I. — Kinematics  of  Machinery 8vo,  i  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4*00 

Velocity  Diagrams 8vo,  i  50 

Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Mbyer's  Descriptive  Geometry.     (7n  preM.) 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.   8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  50 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing. .  12 mo,  oo 

Drafting  Instruments  and  Operations 1 2mo,  25 

Manual  of  Elementary  Projection  Drawing I2mo,  50 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  oo 

Plane  Problems  in  Elementary  Geometry i2mo,  25 

Primary  Geometry 1 2mo,  75 

Elements  of*  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's  Kinematics  and   the  Power  of  Transmission.     (Hermann  and 

Klein.) 8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2n:o,  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective '.  .8vo,  a  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 

ELECTRICITY  AND   PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  . . .  i2mo,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.)     8vo,  300 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  :6mo,  morocco,  ^  oo 
Dolezalek's    Theory   of    the    Lead    Accumulator    (Storage    Battery).     (Von 

Ende.) i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power 121110,  3  oo 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  a  50 

Hanchett's  Alternating  Currents  Explained i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  a  50 

Holman's  Precision  of  Measurements 8vo,  a  oo 

Telescopic  Mirror-scale  Method,  Adjustments,  and  Tests Large  8vo,  75 

Kinzbrunner's  Testing  of  Continuous-Current  Machines 8vo,  2  oo 

Landauer's  Spectrum  Analysis.    (Tingle.) 8vo,  300 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )iamo.  3  oo 

LBb't  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz. )  i  amo,  i  oo 

9 


*  Lyons's  Treatise  on  Electromagnetic  Phenomena.    Vols.  I.  and  H.  Svo,  each,  6  oo 

*  Michie.     Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishoack.) i2mo,  2  50 

*  Rosenberg's  Electrical  Engineering.    (Haldane  Gee — Kinzbrunner.) 8vo,  i  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  L 8vo,  2  50 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  2  oo 

Ulke'js  Modern  Electrolytic  Copper  Refining Svo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Military  Law  ol  United  States Svo,  7  oo 

Sheep,  7  50 

Manual  for  Courts-martial 1 6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence Svo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture      Svo,  5  oo 

Sheep,  5  50 

Law  of  Contracts Svo,  3  oo 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

MANUFACTURES. 

Bernadou's  Smokeless  Powder— Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule. i2mo,  2  50 

Holland's  Iron  Founder I2mo,  2  50 

**  The  Iron  Founder,"  Supplement. 12010,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding I2mo,  3  oo 

Eissler's  Modern  High  Explosives .Svo,  4  oo 

Effront's  Enzymes  and  their  Applications.     ( Prescott. ) Svo  3  oo 

Fitzgerald's  Boston  Machinist iSmo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers iSmo,  i  oo 

Hopkins's  Oil-chemists'  Handbook. „ Svo,  3  oo 

Keep's  Cast  Iron Svo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control.     (In  preparation.) 

Matthews's  The  Textile  Fibres Svo,  3  50 

Metcalf's  SteeL     A  Manual  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures— And  the  Administration   of  Workshops, 

Public  and  Private Svo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories. x6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8.vo,   25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  oo 

Smith's  Press-working  of  Metals Svo,  3  oo 

Spalding's  Hydraulic  Cement izmo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.. .  i6mo  morocco,  2  oo 
Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced.     (In 

press.) 

Thorston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion   Svo,  5  oo 

*  Walke's  Lectures  on  Explosives Svo,  4  oo 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book X2mo,  2  50 

10 


Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 

Wood's  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and  Steel. .  .8vo,  4  oo 

MATHEMATICS. 

Baker's  Elliptic  Function* 8vo,  1  50 

•  Bass's  Elements  of  Differential  Calculus izmo,  4  oo 

Briggs's  Elements  of  Plane  Analytic  Geometry 12 mo,  oo 

Compton's  Manual  of  Logarithmic  Computations 1 2mo,  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo.  50 

•  Dickson's  College  Algebra Large  1 2010,  50 

•  Introduction  to  the  Theory  of  Algebraic  Equations Large  12 mo,  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  50 

Halsted's  Elements  of  Geometry 8vo.  75 

Elementary  Synthetic  Geometry 8vo,  50 

Rational  Geometry tamo, 

•  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:  Vest-pocket  size,  .paper,  15 

100  copies  for  5  oo 

•  Mounted  on  heavy  cardboard,  8  X  to  inches,  25 

10  copies  for  2  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.     Small  8vo,  3  oo 

Johnson'r  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus.  .  Small  8vo,  i   50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates tamo,  i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,  3  50 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  .  12010,  i  50 

•  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  lamo,  2  oo 

•  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,  3  oo 

Trigonometry  and  Tables  published  separately Each,  2  oo 

•  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,  i  oo 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  oo 

Merriman's  Method  of  Least  Squares 8vo,  2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus .  Sm.,  8vo,  3  oo 

Differential  and  Integral  Calculus,     a  vote,  in  one Small  8vo.  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical I2mo,  i  oo     • 

MECHANICAL   ENGINEERING. 

MATERIALS  OF  ENGINEERING.  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice tamo,  50 

Baldwin's  Steam  Heating  for  Buildings lamo,  50 

Barr's  Kinematics  of  Machinery 8vo,  50 

•  Bartlett's  Mechanical  Drawing 8vo,  oo 

•  -                        Abridged  Ed 8vo,  50 

Benjamin's  Wrinkles  and  Recipes xamo,  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings .8vo,  4  oo 

Gary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal     (In  prep- 
aration.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,    paper,  i  oo 

Coolidge  and  Freeman's  Slements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

11 


Cromwell's  Treatise  on  Toothed  Gearing I2mo  i  50 

Treatise  on  Belts  and  Pulleys xamo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flatter's  Dynamometers  and  the  Measurement  of  Power 12 mo,  3  oo 

Rope  Driving I2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo.  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.— Kinematics  of  Machinery ~ 8vo»  I  5<> 

Part  IL— Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shops,  Tools,  and  Methods.     (In  prett.) 

MacCord's  Kinematics;  or,  Practical  Mechanism Svo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

Mahan's  Industrial  Drawing.    (Thompson.) 8vo,  3  So 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo.  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design . .  8vo,  3  oo 

Richards's  Compressed  Air I2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Press-working  of  Metals -   8vo,  3  oo 

Thurston's  Treatise  on  Friction  and    Lost  Work  in   Machinery  and   Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  izmo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 870,  7  So 

Weisbach's  Kinematics  and  the  Power  of  Transmission.      Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) 8vok  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

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Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  SO 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition 

Reset '. 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Johnson'"  Materials  of  Construction Large  8vot  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo.  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

M erriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials   I2mo,  i  oo 

Metcalf's  SteeL     A  Manual  for  Steel-users i2mo  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines i2mo.  i  oo 

Thurston's  Materials  of  Engineering 3  vote..  Svo.  8  oo 

Part   n.— Iron  and  Steel Svo,  3  5° 

Part  HI. — A  Treatise  on  Brasses.  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo  2  50 

Text-book  of  the  Materials  of  Construction Svo,  5  oo 

12 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and  Steel 

8vo,   4  oo 


STEAM-ENGINES  AND  BOILERS. 

Carnot's  Reflections  on  the  Motive  Power  of  He*t.     (Thurrton.) lamo,  i  50 

Dawson's  "Engineering**  and  Electric  Traction  Pocket-book. .  <6mo,  mcr.,  5  oo 

Ford's  Boiier  Makir^  for  BoUer  Makers i8mo,  i  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hemrnway's  Indicator  Practce  and  Steam-engine  Economy lamo,  2  oo 

Hutton'*  Mechanical  Engineering  of  Power  Plants 8vo,  5  oo 

Heat  and  Heat-engines 8vo,  5  co 

Kent's  Steam-boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to.  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator lamo,  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engine* 8vo,  5  oo 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo.  4  oo 

Pr«y*s  Twenty  Years  with  the  Indicator Large  &vo.  2  50 

Pupln's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors 

(Osterberg.) lamo.  I   25 

Reagan's  Locomotives :  Simple,  Compound,  and  Electric iamo.  2  50 

Rontgen's  Principles  of  Thermodynamics.    (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management zamo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice lamo,  a  50 

Snow's  Steam-boiler  Practice 8vo,  3  oo 

Spangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics iamo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo,  i   50 

Manual  of  the  Steam-engine a  vole.  8vo,  10  oo 

Part  I.— History,  Structuce,  and  Theory 8vo,  6  oo,. 

Part  II. — Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice iamo,  i  50 

Manual  of  Steam-boiler* ,  Their  Designs,  Construction,  and  Operation . 8vo,  5  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) x6mo,  a  50 

Wood's  Thermodynamics  Heat  Motors,  and  Refrigerating  Machines 8vo,  4  oo 


MECHANICS    AND  MACHINERY. 

Bart's  Kinematics  of  Machinery 8vo,  a  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making zamo,  a  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

13 


Church's  Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal-working i2mo,  50 

Compton  and  De  Groodt's  The  Speed  Lathe 12010.  50 

Cromwell's  Treatise  on  Toothed  Gearing 12010,  50 

Treatise  on  Belts  and  Pulleys i2mo,  50 

Dana's  Text-book  of  Elementary  Mechanics  for  the  Use  of  Colleges  and 

SchOOlS I2H10,  50 

Dingey's  Machinery  Pattern  Making izmo.  oo 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to   half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

VoL     I.— Kinematics 8vo,  3  50 

Vol.    n. — Statics   8vo.  4  oo 

Vol.  III.— Kinetics 8vo,  3  50 

Mechanics  of  Engineering.     Vol.   I Small   4to.  7  50 

VoL  IL Small  410,  i o  oo 

Durley's  Kinematics  of  Machines  8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo.  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12  mo,  3  oo 

Rope  Driving I2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part  I. — Kinematics  of  Machinery 8vo,  i  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kefir's  Power  and  Power  Transmission 8vo.  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard  s  Machine  Shops,  Tools,  and  Methods.    (In  press.) 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics. 8vo,  4  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo .  4  oo 

Elements  of  Mechanics I2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo  4  oo 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric 12 mo,-  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design . .  8vo,  3  oo 

Richards's  Compressed  Air I2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.    Vol.  I . '. 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Materials  of  Machines. izmo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in  Machinery  and  Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  1 2mo ,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the  Power  of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.    (Herrmann— Klein.). 8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics izmo  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

14 


METALLURGY. 

Bgleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

VoL   I.  —  Silver  ..............................................  STO,  7  50 

VoL   II.  —  Gold  and  Mercury  ...................................  8vo,  7  50 

**  Iles's  Lead-smelting.    (Postage  9  cents  additional.)  .............  xamo.  2  50 

Keep's  Cast  Iron  .................................................  STO,  2  50 

Kuunardt's  Practice  of  Ore  Dressing  in  Europe  ......................  8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard  —  Burgess.)  .  izmo,  3  oo 

Metcalf'  s  SteeL     A  Manual  for  Steel-users  ..........................  zamo,  2  oo 

Smith's  Materials  of  Machines  ....................................  tamo,  i  oo 

Thurston's  Materials  of  Engineering.    In  Three  Parts  ................  8vo,  8  oo 

Part   n.—  Iron  and  Steel  ......................................  8vo,  3  50 

Part  IIL—  A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and   their 

Constituents  ...........................................  8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining  ..........................  8vo,  3  oo 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.     Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia  .............................  STO.  3  oo 

Map  of  Southwest  Virginia  .........................  Pocket-book  form,  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.)  ............  8vo,  4  oo 

Chester's  Catalogue  of  Minerals  ..............................  8vo,  paper,  i  oo 

•                   Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals  ............................  8vo,  3  50 

Dana's  System  of  Mineralogy  .....................  Large  8vo,  half  leather,    12  50 

First  Appendix  to  Dana's  New  "System  of  Mineralogy.'*  ____  Large  8  vo,  i  oo 

Text-book  of  Mineralogy  ......................................  8vo,  4  oo 

Minerals  and  How  to  Study  Them  ............................  12  mo,  i  50 

Catalogue  of  American  Localities  of  Minerals  ..............  Large  8vo,  i  oo 

M^nqai  of  Mineralogy  and  Petrography  .......................  i2tno,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects  ..............  i2mo,  i  oo 

Eakle's  Mineral  Tables.  ...........................................  8vo,  i   25 

£gleston's  Catalogue  of  Minerals  and  Synonyms  .....................  8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.)  Small  8vo,  2  oo 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses.  ............  STO,  4  oo 

•  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  o  50 
Rosenbusch's   Microscopical  Physiography  of  the   Rock-making   Minerals. 

(Iddings.)  ...............................................  8vo,  5  oo 

•  Tillman's  Text-book  of  Important  Minerals  and  Docks  ...............  8vo,  2  oo 

WUliams's  Manual  of  Lithology  ....................................  STO,  3  oo 


Beard's  Ventilation  of  Mines  .....................................  xamo,  2  50 

Boyd's  Resources  of  Southwest  Virginia  .............................  STO,  3  oo 

Map  of  Southwest  Virginia  ........................  Pocket-book  form,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects  ..............  i2mo,  i  oo 

•  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills. 

4to,  half  morocco,    25  oo 

Eissler's  Modern  High  Explosives  ..................................  8vo,  4  oo 

Fowler's  Sewage  Works  Analyses  .................................  xarno,  2  oo 

Goodyear  's  Coal-mines  of  the  Western  Coast  of  the  United  States  ......  lamo,  2  50 

Ihlseng's  Manual  of  Mining  .......................................  STO,  4  oo 

**  Iles's  Lead-smelting.    (Postage  oc.  additional.)  ..................  xamo,  a  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe  .......................  STO,  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores  .....................  STO,  2  oo 

*  Walke's  Lectures  on  Explosives  ..................................  STO,  4  oo 

Wilson's  Cyanide  Processes  ......................................  xamo,  x  50 

Chlorination  Process  ........................................  xamo,  i  50 

15 


Wilson's  Hydraulic  and  Placer  Mining I2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation 12010,  i  25 

SANITARY  SCIENCE. 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.). ....  .8vo,  3  oo 

Water-supply  Engineering 8vo,  4  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works I2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economical  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control. 8vo,  7  50 

Mason's  Water-supply.     (Considered    Principally    from    a    Sanitary    Stand- 
point.)    3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  oo 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Reference 

to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation I2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science ' i2mo,  i  oo 

Richards    and  Woodman's  Air,  Water,  and  Food    from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer.    8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.).  . ,  , 12 mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  and  Military  Hygiene i6mo,  i  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff.) i2mo,  2  50 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,  4  oo 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  Small  8vo,  3  oo 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  i  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  oo 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance 
and  Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small 

Hospital. i2mo,  i  25 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Grammar  of  the  Hebrew  Language 8vo,  3  oo 

Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oo 

Letteris's  Hebrew  Bible 8v°,  2  25 

16 


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